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CHAPTER 2 2.1
LIMITS AND CONTINUITY
RATES OF CHANGE AND TANGENTS TO CURVES
1. (a)
f x
f (3) f (2) 3 2
2. (a)
g x
g (3) g (1) 3 1
3. (a)
h t
g t
4. (a)
h
2819 19
(b)
f x
f (1) f ( 1) 1( 1)
3 ( 1) 2
2
(b)
g x
g (4) g ( 2) 4 ( 2)
34 h 4 11 4
(b)
h t
g t
3 4
4
2
g ( ) g (0) 0
(2 1) (2 1) 0
2
5.
R
R (2) R (0) 20
812 1 321 1
6.
P
P (2) P (1) 21
7. (a) (b) 8. (a) (b)
(816 10) (1 4 5) 1
y x
((2 h )2 5) (22 5) h
y x
(7(2 h )2 ) (7 22 ) h
(b)
h
22 0 1 8 6 8 0
2 h 6 0 2
6
g ( ) g ( ) ( )
3
3
3 3
(2 1) (2 1) 2
0
22 0
2 2 4 4h hh 51 4h h h 4 h. As h 0, 4 h 4 at P (2, 1) the slope is 4. y ( 1) 4( x 2) y 1 4 x 8 y 4 x 9
2 2 744hh h 3 4 hhh 4 h. As h 0, 4 h 4 at P (2, 3) the slope
is 4. y 3 ( 4)( x 2) y 3 4 x 8 y 4 x 11 y
((2 h )2 2(2 h ) 3) (22 2(2) 3)
y
((1 h ) 2 4(1 h )) (12 4(1))
y
(2 h )3 23
y
2 (1 h )3 (2 13 )
4 4 h h 2 4 2 h 3( 3)
9. (a) x h h P(2, 3) the slope is 2. (b) y (3) 2( x 2) y 3 2 x 4 y 2 x 7. 1 2 h h 2 4 4 h ( 3)
2 2h h h 2 h. As h 0, 2 h 2 at
2
10. (a) x h h 2h h 2. As h 0, h 2 2 at P (1, 3) the h h slope is 2. (b) y ( 3) ( 2)( x 1) y 3 2 x 2 y 2 x 1. 2
3
2
3
11. (a) x 812h 4hh h 8 12 h 4hh h 12 4h h 2 . As h 0, 12 4h h 2 12, at P (2, 8) h the slope is 12. (b) y 8 12( x 2) y 8 12 x 24 y 12 x 16. 2
3
2
3
12. (a) x 213h h3h h 1 3h 3hh h 3 3h h 2 . As h 0, 3 3h h 2 3, at h P(1, 1) the slope is 3. (b) y 1 ( 3)( x 1) y 1 3 x 3 y 3x 4.
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62
Chapter 2 Limits and Continuity 3
3
2
3
2 3 y (1 h ) 12(1 h ) (1 12(1)) 13h 3h h 1212 h ( 11) 13. (a) x 9h 3hh h 9 3h h 2 . h h As h 0, 9 3h h 2 9 at P(1, 11) the slope is 9. (b) y (11) (9)( x 1) y 11 9 x 9 y 9 x 2.
(2 h )3 3(2 h ) 2 4 (23 3(2)2 4)
y
2
3
12 h 3h 14. (a) x 812h 6h h 12 h h 2 As h 0, 3h h 0 at P (2, 0) the slope is 0. (b) y 0 0( x 2) y 0.
15. (a)
y x
1 1 2 h 2
h
As h 0, (b)
16. (a)
17. (a)
1 2( 2 h )
y x
(4 h ) 4 2 (4 h ) 2 4
h 1 2 h
18. (a)
y x
41 , at P 2, 21 the slope is 41 .
4 h 2( 2 h ) 1 42hh 12 1h h 21h 21 h . 2 h 1, 2
at P (4, 2) the slope is
1. 2
(4 h ) 4 1 4 hh 4 4hh 2 4 h 2 . 4 h 2 h ( 4 h 2) 4 h 2
1 4 h 2
1 4 2
14 , at P (4, 2) the slope is
1. 4
y 2 14 ( x 4) y 2 14 x 1 y 14 x 1 y x
7 ( 2 h ) 7 ( 2) h
As h 0, (b)
2 3 3h h h 3h h 2 .
y (2) 12 ( x 4) y 2 12 x 2 y 12 x 4
As h 0, (b)
40
y 21 41 ( x (2)) y 12 41 x 12 y 41 x 1
As h 0, (b)
2 ( 2 h ) 2( 2 h ) 1h 2( 21 h) .
2
1 9 h 3
9 h 3 h
9 h 3 9 h 3 h 9 h 3
(9 h ) 9 h ( 9 h 3)
1 . 9 h 3
1 61 , at P (2, 3) the slope is 61 . 9 3
y 3 61 ( x (2)) y 3 61 x 13 y 61 x 83
19. (a)
Q Q1 (10, 225)
Q2 (14,375) Q3 (16.5, 475) Q4 (18,550)
Slope of PQ 650 225 2010 650 375 20 14 650 475 20 16.5 650 550 2018
p t
42.5 m/sec 45.83 m/sec 50.00 m/sec 50.00 m/sec
(b) At t 20, the sportscar was traveling approximately 50 m/sec or 180 km/h.
20. (a)
Q Q1 (5, 20) Q2 (7, 39)
Q3 (8.5,58) Q4 (9.5, 72)
p
Slope of PQ t 80 20 12 m/sec 10 5
80 39 13.7 m/sec 10 7 80 58 14.7 m/sec 10 8.5 8072 16 m/sec 10 9.5
(b) Approximately 16 m/sec
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Section 2.1 Rates of Change and Tangents to Curves 21. (a)
63
p
Profit (1000s)
200 160 120 80 40 0
(b)
p t
2010 2011 2012 2013 2014 Ye ar
174 62 2014 2012
t
112 56 thousand dollars per year 2 p
(c) The average rate of change from 2011 to 2012 is t
62 27 20122011
35 thousand dollars per year.
p
11162 49 thousand dollars per year. The average rate of change from 2012 to 2013 is t 2013 2012 So, the rate at which profits were changing in 2012 is approximately 12 (35 49) 42 thousand dollars per year.
22. (a) F ( x) ( x 2)/( x 2) x 1.2 1.1 F ( x) 4.0 3.4 F x F x F x
1.01 3.04
1.001 3.004
4.0 ( 3) 5.0; 1.2 1 3.04 ( 3) 4.04; 1.011 3.0004 ( 3) 4.0004; 1.00011
F x F x
g x
1.0001 3.0004
1 3
3.4 ( 3) 4.4; 1.11 3.004 ( 3) 4.004; 1.0011
(b) The rate of change of F ( x) at x 1 is 4. g
23. (a) x g x
g (2) g (1) 2211 0.414213 2 1 g (1 h ) g (1) 1hh 1 (1 h ) 1
g (1.5) g (1) 1.51
1.5 1 0.5
0.449489
(b) g ( x) x 1 h
1 h
1 h 1 /h
1.1
1.01
1.001
1.0001
1.00001
1.000001
1.04880
1.004987
1.0004998
1.0000499
1.000005
1.0000005
0.4880
0.4987
0.4998
0.499
0.5
0.5
(c) The rate of change of g ( x) at x 1 is 0.5. (d) The calculator gives lim
h 0
24. (a) i) ii)
1 h 1 h
12 .
11 1 f (3) f (2) 3 1 2 16 16 3 2 1 1 2 T f (T ) f (2) TT 22 2TT 22T 2T2(TT 2) T 2
T 2T2(2 21T , T 2 T )
(b) T f (T ) ( f (T ) f (2))/(T 2)
2.1 2.01 2.001 0.476190 0.497512 0.499750 0.2381 0.2488 0.2500 (c) The table indicates the rate of change is 0.25 at t 2. (d) lim 21T 14 T 2
2.0001 0.4999750 0.2500
2.00001 0.499997 0.2500
NOTE: Answers will vary in Exercises 25 and 26. 0 15 mph; [1, 2.5]: s 25. (a) [0, 1]: st 15 t 10
20 15 2.51
30 20 10 mph 10 mph; [2.5, 3.5]: st 3.5 2.5 3
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2.000001 0.499999 0.2500
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64
Chapter 2 Limits and Continuity (b) At P
12 , 7.5 : Since the portion of the graph from t 0 to t 1 is nearly linear, the instantaneous rate of
change will be almost the same as the average rate of change, thus the instantaneous speed at t 12 is 157.5 15 mi/hr. At P (2, 20): Since the portion of the graph from t 2 to t 2.5 is nearly linear, the 10.5 20 0 mi/hr. instantaneous rate of change will be nearly the same as the average rate of change, thus v 20 2.5 2 For values of t less than 2, we have Q Q1 (1, 15)
Q2 (1.5, 19) Q3 (1.9, 19.9)
Slope of PQ
s t
15 20 5 mi/hr 1 2 19 20 2 mi/hr 1.5 2 19.9 20 1 mi/hr 1.9 2
Thus, it appears that the instantaneous speed at t 2 is 0 mi/hr. At P(3, 22): Q Q1 (4, 35) Q2 (3.5, 30) Q3 (3.1, 23)
Slope of PQ 35 22 4 3 30 22 3.53 23 22 3.13
s t
Q
13 mi/hr
Q1 (2, 20)
16 mi/hr
Q2 (2.5, 20)
10 mi/hr
Q3 (2.9, 21.6)
Slope of PQ
s t
20 22 2 mi/hr 2 3 20 22 4 mi/hr 2.53 21.6 22 4 mi/hr 2.9 3
Thus, it appears that the instantaneous speed at t 3 is about 7 mi/hr. (c) It appears that the curve is increasing the fastest at t 3.5. Thus for P(3.5, 30) Slope of PQ st Slope of PQ Q Q Q1 (4, 35) Q2 (3.75, 34)
Q3 (3.6, 32)
3530 10 mi/hr 43.5 34 30 16 mi/hr 3.753.5 32 30 20 mi/hr 3.6 3.5
Q1 (3, 22) Q2 (3.25, 25) Q3 (3.4, 28)
s t
2230 16 mi/hr 33.5 2530 20 mi/hr 3.253.5 2830 20 mi/hr 3.4 3.5
Thus, it appears that the instantaneous speed at t 3.5 is about 20 mi/hr. gal
1.67 day ; [0, 5]: At 26. (a) [0, 3]: At 10315 0
(b) At P(1, 14): Q Q1 (2, 12.2) Q2 (1.5, 13.2) Q3 (1.1, 13.85)
Slope of PQ
A t
12.2 14 1.8 gal/day 21 13.2 14 1.6 gal/day 1.51 13.8514 1.5 gal/day 1.11
3.915 5 0
gal
gal
2.2 day ; [7, 10]: At 0101.4 0.5 day 7
Q Q1 (0, 15) Q2 (0.5, 14.6)
Q3 (0.9, 14.86)
Slope of PQ
A t
1514 1 gal/day 0 1 14.6 14 1.2 gal/day 0.51 14.86 14 1.4 gal/day 0.9 1
Thus, it appears that the instantaneous rate of consumption at t 1 is about 1.45 gal/day. At P(4, 6): Slope of PQ At Q Slope of PQ A Q t
Q1 (5, 3.9)
Q2 (4.5, 4.8) Q3 (4.1, 5.7)
3.9 6 5 4 4.86 4.5 4 5.7 6 4.1 4
2.1 gal/day
Q1 (3, 10)
2.4 gal/day
Q2 (3.5, 7.8)
3 gal/day
Q3 (3.9, 6.3)
10 6 4 gal/day 3 4 7.86 3.6 gal/day 3.5 4 6.36 3 gal/day 3.9 4
Thus, it appears that the instantaneous rate of consumption at t 1 is 3 gal/day. (solution continues on next page)
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Section 2.2 Limit of a Function and Limit Laws At P(8, 1): Q Q1 (9, 0.5)
Q2 (8.5, 0.7) Q3 (8.1, 0.95)
Slope of PQ
Q
A t
Q1 (7, 1.4)
0.51 0.5 gal/day 9 8 0.7 1 0.6 gal/day 8.58 0.951 0.5 gal/day 8.18
Q2 (7.5, 1.3) Q3 (7.9, 1.04)
Slope of PQ
65
A t
1.4 1 0.6 gal/day 7 8 1.31 0.6 gal/day 7.58 1.04 1 0.6 gal/day 7.98
Thus, it appears that the instantaneous rate of consumption at t 1 is 0.55 gal/day. (c) It appears that the curve (the consumption) is decreasing the fastest at t 3.5. Thus for P(3.5, 7.8) Slope of PQ st Q Slope of PQ At Q 11.2 7.8 3.4 gal/day Q1 (2.5, 11.2) 4.8 7.8 3 gal/day Q1 (4.5, 4.8) 2.53.5 Q2 (4, 6)
Q3 (3.6, 7.4)
4.53.5 6 7.8 3.6 gal/day 4 3.5 7.4 7.8 4 gal/day 3.6 3.5
Q2 (3, 10)
Q3 (3.4, 8.2)
10 7.8 4.4 gal/day 33.5 8.2 7.8 4 gal/day 3.4 3.5
Thus, it appears that the rate of consumption at t 3.5 is about 4 gal/day. 2.2
LIMIT OF A FUNCTION AND LIMIT LAWS
1. (a) Does not exist. As x approaches 1 from the right, g ( x) approaches 0. As x approaches 1 from the left, g ( x) approaches 1. There is no single number L that all the values g ( x) get arbitrarily close to as x 1. (b) 1 (c) 0 (d) 0.5 2. (a) 0 (b) 1 (c) Does not exist. As t approaches 0 from the left, f (t ) approaches 1. As t approaches 0 from the right, f (t ) approaches 1. There is no single number L that f (t ) gets arbitrarily close to as t 0. (d) 1 3. (a) (d) (g) (j)
True False True True
(b) (e) (h) (k)
4. (a) False (d) True (g) False x x0 | x |
(c) False (f) True (i) True
(b) False (e) True (h) True
(c) True (f) True (i) False
1 if x 0 and | xx | xx 1 if x 0. As x approaches 0 from the left, | xx | approaches 1. As x approaches 0 from the right, | xx | approaches 1. There is no single number L that all the
5. lim
does not exist because | xx |
True False False False
x x
function values get arbitrarily close to as x 0.
1 become increasingly large and negative. As x approaches 1 6. As x approaches 1 from the left, the values of x 1 from the right, the values become increasingly large and positive. There is no number L that all the function values get arbitrarily close to as x 1, so lim x11 does not exist. x 1
7. Nothing can be said about f ( x) because the existence of a limit as x x0 does not depend on how the function is defined at x0 . In order for a limit to exist, f ( x) must be arbitrarily close to a single real number L when x is close enough to x0 . That is, the existence of a limit depends on the values of f ( x) for x near x0 , not on the definition of f ( x) at x0 itself. Copyright 2018 Pearson Education, Inc.
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66
Chapter 2 Limits and Continuity
8. Nothing can be said. In order for lim f ( x) to exist, f ( x) must close to a single value for x near 0 regardless of x 0
the value f (0) itself.
9. No, the definition does not require that f be defined at x 1 in order for a limiting value to exist there. If f (1) is defined, it can be any real number, so we can conclude nothing about f (1) from lim f ( x) 5. x 1
10. No, because the existence of a limit depends on the values of f ( x) when x is near 1, not on f (1) itself. If lim f ( x) exists, its value may be some number other than f (1) 5. We can conclude nothing about lim f ( x), x 1
x 1
whether it exists or what its value is if it does exist, from knowing the value of f (1) alone. 11.
lim ( x 2 13) ( 3)2 13 9 13 4
x3
12. lim ( x 2 5 x 2) (2)2 5(2) 2 4 10 2 4 x 2
13. lim 8(t 5)(t 7) 8(6 5)(6 7) 8 t 6
14.
15.
16.
17.
lim ( x3 2 x 2 4 x 8) (2)3 2(2)2 4( 2) 8 8 8 8 8 16
x 2
lim 2 x 53 x2 11 x
2(2) 5 11(2)3
lim (8 3s )(2 s 1) 8 5
t 2/3
y2
2 y 2 y 5 y 6
20.
2 2 (2) 2 5(2) 6
2
( 2) 23 4
y 3
lim
z 4
22. lim
5h 4 2 h
h 0
x 5 2 x 5 x 25
23. lim
2
2 ( 2) 25 25 2
4
24 16
z 2 10 42 10 16 10 6 3 3h 1 1
h 0
4 4 1 410 20 6 5
lim (5 y ) 4/3 [5 (3)]4/3 (8)4/3 (8)1/3
21. lim
24.
23 2 23 1 (8 2) 43 1 (6) 13 2
lim 4 x (3x 4)2 4 12 3 12 4 x 1/2
18. lim
19.
93 3
3 3(0) 1 1
lim
h 0
3 32 1 1
5 h 4 2 5h 4 2 h 5h 4 2
lim
h 0 h
(5h 4) 4
5h 4 2
lim
h 0 h
5h 5h 4 2
lim
h0
5 5h 4 2
x 5 lim ( x 5)( lim 1 1 1 x 5) x 5 x 5 55 10 x 5
lim 2 x 3 x 3 x 4 x 3
x 3 lim ( x 3)( lim 1 1 12 x 1) x 3 x 1 31 x 3
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5 4 2
54
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Section 2.2 Limit of a Function and Limit Laws 25.
2 ( x 5)( x 2) lim x x3x510 lim lim ( x 2) 5 2 7 x 5 x 5 x 5
x 5
x 2 7 x 10 x2 x 2
26. lim
t 2 t 2 2 t 1 t 1
27. lim
( x 5)( x 2) x2
lim
x2
lim ( x 5) 2 5 3 x 2
(t 2)(t 1)
lim (t 1)(t 1) lim tt12 1112 32 t 1 t 1
28.
2 (t 2)(t 1) lim t 23t 2 lim (t 2)(t 1) lim tt 22 11 22 13 t t 2 t 1 t 1 t 1
29.
lim 32 x 42 x 2 x 2 x
30. lim
2( x 2)
lim
lim 22 42 12 x 2 x
2 x 2 x ( x 2)
5 y 3 8 y 2
y 2 (5 y 8)
lim
4 2 y 0 3 y 16 y
816 12
2 y 0 3 y 16
1 x
5 y 8
lim
2 2 y 0 y (3 y 16)
31.
1 lim xx 11 lim x x 1 lim 1xx x11 lim 1x 1 x1 x 1 x1 x1
32.
lim
x 0
1 1 x 1 x 1
x
u 4 1 3 u 1 u 1
33. lim
lim
lim
4 x x2 x 4 2 x
lim
x 1
lim
x 1
x 1 x 3 2
lim
x 2 8 3 x 1
x 1
( x 1)
x 3 2
lim
x 1
lim
39. lim
x 2
x 2 12 4 x 2
( x 1)
x 1
x 2 12 4
lim
x2
x2
x 2 12 4
x 1
x 2 8 3
lim
lim
x 8 3
x 2 8 3
( x 2)
2
x 1
x 2
x 3 2
x 2 8 3
1 9 3
x (2 x )(2 x ) 2 x
x 3 2
1 x 3
x 9
x 4
(11)(11) 111
v 2 2v 4 2 v 2 (v 2)( v 4)
lim
lim
2 x 0 ( x 1)( x 1)
lim
v 2 (v 2)(v 2)( v 4) x 3 x 9 ( x 3)( x 3)
lim
u 2 u 1
u 1
2
x (4 x ) x 4 2 x
(u 2 1)(u 1)
lim
(v 2)( v 2 2v 4)
lim x 3 x 9 x 9
37. lim
38.
2 u 1 (u u 1)(u 1)
lim
2x 1 x 0 ( x 1)( x 1) x
lim
(u 2 1)(u 1)(u 1)
v3 8 4 v 2 v 16
36. lim
x
x 0
34. lim
35.
( x 1) ( x 1) ( x 1)( x 1)
lim
2
2 3 3
2 1
2
4 3
12 3 32 8
16
x 3 2
lim
lim x 2 x 4(2 2) 16 x 4
( x 1)
( x 3) 4
lim
x 1
( x 2 8) 9
x 1 ( x 1)
x 2 8 3
x32 4 24
lim
( x 1)( x 1)
x 1 ( x 1)
x 2 8 3
13
x 2 12 4
x 12 4
4 4 4 (4)(8)
4 16 4
lim
( x 2 12) 16
x 2 ( x 2)
1 2
2
x 12 4
lim
( x 2)( x 2)
x 2 ( x 2)
x 2 12 4
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68 40.
Chapter 2 Limits and Continuity x2
lim
x 2
2
x 5 3
( x 2)
lim
x 2
x 2
2 lim 2 x x3 5 x 3
lim
x 2 5 3
x 5 3
9 3 4
x 5 3
2
x 3
2
2
(3 x )(3 x )
4 x 2
x 4 5 x 9
lim
(4 x ) 5 x 2 9
(4 x ) 5 x 2 9 (4 x )(4 x )
x 4
x 2
1 x 0 cos x
47. lim
x 0
1 x sin x 3cos x
1 cos 0
lim
( x 2)
x 2 5 3
( x 2)( x 2)
x 2
3 x x 3 2 x 2 5
x 3 ( x 3) 2 x 5
lim
x4
lim 5 x 4
2
6 2 4
32
(4 x ) 5 x 2 9 2
25( x 9)
x2 9 4 x
5 25 8
9 x 2
lim
x 3 ( x 3) 2 x 2 5
lim (4 x)5 x4
16 x
x 2 9 2
5 4 2
44.
x 0
x 0
x 2 5 3
4 ( x 2 5)
43. lim (2sin x 1) 2 sin 0 1 0 1 1 45. lim sec x lim
( x 2 5) 9
lim
x 4 5 x 2 9 5 x 2 9
lim
lim
lim
x 2 5 2 x 2 5
x 3 ( x 3) 2 x 2 5
42. lim
( x 2)
32
( x 3) 2 x 5
lim
2
x 2 5 3 x 2
lim
41.
11 1
46.
lim sin 2 x lim sin x (sin 0) 2 02 0 x 0 x 0 sin x x 0 cos x
lim tan x lim
x 0
sin 0 cos 0
0 1
0
0 sin 0 1 0 0 1 13cos 0 3 3
48. lim ( x 2 1)(2 cos x) (02 1)(2 cos 0) (1)(2 1) (1)(1) 1 x 0
49.
lim
x
x 4 cos( x ) lim
50. lim 7 sec2 x x 0
x 4 lim cos( x ) 4 cos 0 4 1 4
x
x
lim (7 sec2 x) 7 lim sec2 x 7 sec2 0 7 (1) 2 2 2
x 0
x 0
51. (a) quotient rule (c) sum and constant multiple rules
(b) difference and power rules
52. (a) quotient rule (c) difference and constant multiple rules
(b)
power and product rules
53. (a) lim f ( x) g ( x) lim f ( x) lim g ( x) (5)(2) 10 x c x c x c (b) lim 2 f ( x) g ( x) 2 lim f ( x ) lim g ( x) 2(5)( 2) 20 x c x c x c (c) lim [ f ( x ) 3 g ( x)] lim f ( x) 3 lim g ( x) 5 3( 2) 1 x c
(d)
f ( x) lim x c f ( x ) g ( x )
x c lim f ( x )
xc
lim f ( x ) lim g ( x )
x c
x c
x c
5 5 5( 2) 7
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Section 2.2 Limit of a Function and Limit Laws
69
54. (a) lim [ g ( x) 3] lim g ( x) lim 3 3 3 0 x 4
x4
x 4
(b) lim xf ( x ) lim x lim f ( x) (4)(0) 0 x 4
x4
x4
2
(c) lim [ g ( x)]2 lim g ( x) [3]2 9 x 4 x 4 g ( x) x 4 f ( x ) 1
(d) lim
lim g ( x )
x 4
lim f ( x ) lim 1
x4
x 4
3 0 1
3
55. (a) lim [ f ( x) g ( x )] lim f ( x) lim g ( x ) 7 (3) 4 x b
x b
x b
(b) lim f ( x) g ( x) lim f ( x ) lim g ( x) (7)(3) 21 x b x b xb (c) lim 4 g ( x) lim 4 lim g ( x) (4)(3) 12 x b x b x b (d) lim f ( x)/g ( x) lim f ( x)/ lim g ( x) 73 73 x b
56. (a)
x b
x b
lim [ p ( x) r ( x) s ( x)] lim p ( x) lim r ( x) lim s ( x ) 4 0 ( 3) 1
x 2
x 2
x 2
x 2
(b) lim p ( x) r ( x) s ( x ) lim p ( x ) lim r ( x) lim s ( x) (4)(0)(3) 0 x 2 x 2 x2 x 2 (c) lim [4 p ( x) 5r ( x)]/s ( x) 4 lim p ( x) 5 lim r ( x ) lim s ( x) [4(4) 5(0)]/ 3 16 3 x 2 x 2 x2 x2 (1 h ) 2 12 h h 0
57. lim 58. lim
h 0
1 2 h h 2 1 h h 0
lim
2
( 2 h ) ( 2) h
2
lim
h 0
[3(2 h ) 4][3(2) 4] h h 0
59. lim
60. lim
h 0
61. lim
h 0
21 h 12 lim 7h 7 h
lim
3h h 0 h
lim
2 1 2 h
h 0
h
63. lim
x 0
h 0
2 ( 2 h )
7 h 7
7 h 7
h 0
h ( h 4) h
lim
h0 2 h ( 2 h )
7h 7
lim (2 h) 2 lim (h 4) 4 h 0
3
lim
3(0 h ) 1 3(0) 1 lim h h 0 h 0
62. lim
h 0
4 4h h2 4 h
h 0 2 h
h
h (2 h ) h
lim
lim
h 0 h
3h 1 1 h
h h 0 h (4 2 h )
(7 h ) 7 7h 7
lim
3h 1 1
3h 1 1
5 2 x 2 5 2(0)2 5 and lim
x0
14
lim
h 0 h
lim
h 0 h
(3h 1) 1
3h 1 1
h 7h 7
lim
h 0 h
lim
h0
3h 3h 1 1
1 7h 7
lim
h0
3 3h11
2
x 0
65. (a) lim 1 x6 1 06 1 and lim 1 1; by the sandwich theorem, lim x0
x0
x 0
x 0
1 2 7
3 2
5 x 2 5 (0) 2 5; by the sandwich theorem, lim f ( x) 5
64. lim (2 x 2 ) 2 0 2 and lim 2 cos x 2(1) 2; by the sandwich theorem, lim g ( x) 2 x 0
x sin x x 0 2 2 cos x
1
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70
Chapter 2 Limits and Continuity (b) For x 0, y ( x sin x)/(2 2 cos x) lies between the other two graphs in the figure, and the graphs converge as x 0.
1 x 0 2
66. (a) lim
x2 24
lim
1 x 0 2
x2 24 x 0
lim
1 2
0
1 2
1 x0 2
and lim
(b) For all x 0, the graph of f ( x) (1 cos x)/x 2 lies between the line y 12 and the parabola
12 ; by the sandwich theorem, lim
x 0
1cos x x2
y 12 x 2 /24, and the graphs converge as
x 0.
67. (a)
f ( x) ( x 2 9)/( x 3)
x
3.1
3.01
3.001
3.0001
3.00001
3.000001
f ( x)
6.1
6.01
6.001
6.0001
6.00001
6.000001
x
2.9
2.99
2.999
2.9999
2.99999
2.999999
f ( x)
5.9
5.99
5.999
5.9999
5.99999
5.999999
The estimate is lim f ( x) 6. x 3
(b)
(c) f ( x)
x 2 9 x 3
( x 3)( x 3) x 3
68. (a) g ( x) ( x 2 2)/ x 2
x g ( x)
1.4 2.81421
x 3 if x 3, and lim ( x 3) 3 3 6. x 3
1.41 2.82421
1.414 2.82821
1.4142 2.828413
1.41421 2.828423
1.414213 2.828426
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12 .
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Section 2.2 Limit of a Function and Limit Laws (b)
(c) g ( x)
x2 2 x 2
x 2 x 2 x x 2
69. (a) G ( x) ( x 6)/( x 2 4 x 12) x 5.9 5.99 G ( x ) .126582 .1251564
x G ( x)
6.1 .123456
6.01 .124843
2 if x 2, and lim
x 2
x 2
2 2 2 2.
5.999 .1250156
5.9999 .1250015
5.99999 5.999999 .1250001 .1250000
6.001 .124984
6.0001 .124998
6.00001 .124999
6.000001 .124999
(b)
(c) G ( x)
x 6 ( x 2 4 x 12)
x6 ( x 6)( x 2)
1 x2
1 x 6 x 2
if x 6, and lim
1 6 2
18 0.125.
70. (a) h( x) ( x 2 2 x 3)/( x 2 4 x 3)
x h( x )
2.9 2.052631
2.99 2.005025
2.999 2.000500
2.9999 2.000050
2.99999 2.000005
2.999999 2.0000005
x 3.1 h( x) 1.952380
3.01 1.995024
3.001 1.999500
3.0001 1.999950
3.00001 1.999995
3.000001 1.999999
(b)
(c) h( x)
x 2 2 x 3 x 2 4 x 3
( x 3)( x 1) ( x 3)( x 1)
x 1 if x 1
x 1 x 3 x 1
x 3, and lim
31 31
4 2
2.
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72
Chapter 2 Limits and Continuity
71. (a) f ( x ) ( x 2 1)/(| x | 1)
x f ( x)
1.1 2.1
1.01 2.01
1.001 2.001
1.0001 2.0001
1.00001 2.00001
1.000001 2.000001
x f ( x)
.9 1.9
.99 1.99
.999 1.999
.9999 1.9999
.99999 1.99999
.999999 1.999999
(b)
(c) f ( x )
x 2 1 x 1
( x 1)( x 1) x 1, x 0 and x 1 x 1 , and lim (1 x ) 1 ( 1) 2. ( x 1)( x 1) x 1 ( x 1) 1 x, x 0 and x 1
72. (a) F ( x) ( x 2 3 x 2)/(2 | x |)
x F ( x)
2.1 1.1
2.01 1.01
2.001 1.001
2.0001 1.0001
2.00001 1.00001
2.000001 1.000001
x F ( x)
1.9 .9
1.99 .99
1.999 .999
1.9999 .9999
1.99999 .99999
1.999999 .999999
(b)
(c) F ( x )
x2 3 x 2 2 x
( x 2)( x 1) , x0 2 x , and lim ( x 1) 2 1 1. ( x 2)( x 1) x 2 x 1, x 0 and x 2 2 x
73. (a) g ( ) (sin )/
g ( )
.1 .998334
.1 g ( ) .998334 lim g( ) 1
.01 .999983
.001 .999999
.01 .001 .999983 .999999
.0001 .999999
.00001 .999999
.000001 .999999
.0001 .999999
.00001 .999999
.000001 .999999
0
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Section 2.2 Limit of a Function and Limit Laws (b)
74. (a) G (t ) (1 cos t )/t 2
t G (t )
.1 .499583
.01 .499995
.001 .499999
.0001 .5
t G (t )
.1 .499583
.01 .499995
.001 .499999
.0001 .00001 .000001 .5 .5 .5
.00001 .5
.000001 .5
lim G (t ) 0.5
t 0
(b)
75. (a)
f ( x) x1/(1 x ) x .9 .99 f(x) .348678 .366032
.999 .367695
.9999 .367861
.99999 .367877
.999999 .367879
x f(x)
1.001 .368063
1.0001 .367897
1.00001 .367881
1.000001 .367878
1.1 .385543
1.01 .369711
lim f ( x) 0.36788
x 1
(b)
Graph is NOT TO SCALE. Also, the intersection of the axes is not the origin: the axes intersect at the point (1, 2.71820).
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74
Chapter 2 Limits and Continuity
76. (a)
f ( x) (3x 1)/x x .1 .01 f(x) 1.161231 1.104669
.1 1.040415
x f(x)
.01 1.092599
.001 1.099215
.0001 1.098672
.00001 1.098618
.000001 1.098612
.001 1.098009
.0001 1.098551
.00001 1.098606
.000001 1.098611
lim f ( x) 1.0986
x 1
(b)
77. lim f ( x) exists at those points c where lim x 4 lim x 2 . Thus, c 4 c 2 c 2 (1 c 2 ) 0 c 0, 1, or 1. x c
x c
x c
Moreover, lim f ( x) lim x 2 0 and lim f ( x ) lim f ( x) 1. x 0
x 0
x 1
x 1
78. Nothing can be concluded about the values of f , g , and h at x 2. Yes, f (2) could be 0. Since the conditions of the sandwich theorem are satisfied, lim f ( x ) 5 0. x 2
f ( x ) 5 x 4 x 2
79. 1 lim
80. (a) 1 lim
lim f ( x ) lim 5
x4
lim x lim 2
x 4
f ( x)
2 x 2 x
(b) 1 lim
x4
f ( x)
2 x 2 x
lim f ( x )
x 2
lim x 2
x 2
4 2
lim f ( x )
x 2
4
f ( x) x
f ( x ) 5 81. (a) 0 3 0 lim x 2 x 2 lim f ( x) 5 lim
(b) 0 4 0 lim x 2
x 4
x4
lim x 2
x2
lim f ( x ) 5
lim f ( x) 5 2(1) lim f ( x) 2 5 7. x 4
lim f ( x) 4. x 2
lim 1 lim x 2 x x 2
lim ( x 2) lim x2 x 2 f ( x) 5.
x 2 f ( x ) 5 lim ( x 2) x 2 x 2
2
x 0
f ( x) x2
f ( x) x
f ( x ) 5 x2
f ( x) 2. 12 xlim 2 x
( x 2) lim [ f ( x) 5] x2
lim f ( x) 5 as in part (a).
f ( x) 82. (a) 0 1 0 lim 2 lim x lim x0 x 0 x x 0 That is, lim f ( x) 0.
(b) 0 1 0 lim x 0
x 4
x 2
f ( x) x2
f ( x) x 2 lim 2 x 2 lim f ( x). xlim x 0 0 x0 x
lim x lim f ( x ) x lim x0 x 0 x0 x 2
f ( x) . That x
is, lim
x0
f ( x) x
0.
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Section 2.3 The Precise Definition of a Limit
75
83. (a) lim x sin 1x 0 x 0
(b) 1 sin 1x 1 for x 0: x 0 x x sin 1x x lim x sin 1x 0 by the sandwich theorem; x 0
x 0 x x sin 1x x lim x sin 1x 0 by the sandwich theorem. x 0
84. (a) lim x 2 cos x 0
(b) 1 cos
0 1 x3
1 for x 0 x 1 x3
2
x 2 cos
lim x 2 0.
x 1 x3
2
lim x 2 cos x 0
0 by the sandwich theorem since 1 x3
x 0
Example CAS commands: 85–90. Maple: f : x - (x^4 16)/(x 2); x0 : 2; plot( f (x), x x0-1..x0 1, color black, title "Section 2.2, #85(a)" ); limit( f (x), x x 0 );
In Exercise 87, note that the standard cube root, x^(1/3), is not defined for x<0 in many CASs. This can be overcome in Maple by entering the function as f : x - (surd(x 1, 3) 1)/x. Mathematica: (assigned function and values for x0 and h may vary) Clear[f , x] f[x _]: (x 3 x 2 5x 3)/(x 1)2 x0 1; h 0.1; Plot[f[x],{x, x0 h, x0 h}] Limit[f[x], x x0]
2.3
THE PRECISE DEFINITION OF A LIMIT
1. Step 1: x 5 x 5 5 x 5 Step 2: 5 7 2,or 5 1 4. The value of δ which assures x 5 1 x 7 is the smaller value, 2. Copyright 2018 Pearson Education, Inc.
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76
Chapter 2 Limits and Continuity
2.
Step 1: Step 2:
x 2 x 2 2 x 2 2 1 1, or 2 7 5. The value of which assures x 2 1 x 7 is the smaller value, 1.
Step 1: Step 2:
x (3) x 3 3 x 3 3 72 12 , or 3 12 52 . The value of which assures x (3) 72 x 12 is the smaller value, 12 .
Step 1:
x 32 x 32 32 x 32
3.
4.
Step 2:
32 72 2, or 32 12 1.
The value of which assures x 32 72 x 12 is the smaller value, 1.
5. Step 1: Step 2:
x 12 x 12 12 x 12
12
4 9
1 , or 1 18 2
4 7
1 . 14
The value of which assures x 12 94 x
4 7
1. is the smaller value, 18
6.
Step 1: Step 2:
7. Step 1: Step 2:
x 3 x 3 3 x 3 3 2.7591 0.2409, or 3 3.2391 0.2391. The value of which assures x 3 2.7591 x 3.2391 is the smaller value, 0.2391. x 5 x 5 5 x 5 From the graph, 5 4.9 0.1, or 5 5.1 0.1; thus 0.1 in either case.
8. Step 1: Step 2:
x (3) x 3 3 x 3 From the graph, 3 3.1 0.1, or 3 2.9 0.1; thus 0.1.
9. Step 1: Step 2:
x 1 x 1 1 x 1 9 7 , or 1 25 9 ; thus 7 . From the graph, 1 16 16 16 16 16
10. Step 1: Step 2:
x 3 x 3 3 x 3 From the graph, 3 2.61 0.39, or 3 3.41 0.41; thus 0.39.
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Section 2.3 The Precise Definition of a Limit 11. Step 1: Step 2:
x 2 x 2 2 x 2 From the graph, 2 3 2 3 0.2679, or 2 5 5 2 0.2361; thus 5 2.
12. Step 1:
x (1) x 1 1 x 1
Step 2:
From the graph, 1 thus
5 2 . 2
5 2
5 2 2
0.118 or 1
13. Step 1: Step 2:
x (1) x 1 1 x 1 From the graph, 1 16 79 0.77, or 1 16 9 25
14. Step 1:
x 12 x 12 12 x 12
Step 2:
From the graph, 12 thus 0.00248.
1 2.01
3 2
9 25
2 3 2
0.36; thus
( x 1) 5 0.01 x 4 0.01 0.01 x 4 0.01 3.99 x 4.01 x 4 x 4 4 x 4 0.01.
16. Step 1:
(2 x 2) (6) 0.02 2 x 4 0.02 0.02 2 x 4 0.02 4.02 2 x 3.98 2.01 x 1.99 x (2) x 2 2 x 2 0.01.
17. Step 1:
x 1 1 0.1 0.1 x 1 1 0.1 0.9 x 1 1.1 0.81 x 1 1.21
Step 2:
0.19 x 0.21 x 0 x . Then, 0.19 0.19 or 0.21; thus, 0.19. x 12 0.1 0.1 x 12 0.1 0.4 x 0.6 0.16 x 0.36
18. Step 1: Step 2:
x 14 x 14 Then 14 0.16 0.09
4 x 19 16 15 x 3 or 3 x 15 x 10 x 10 10 x 10. Then 10 3 7, or 10 15 5; thus 5.
x 7 4 1 1 x 7 4 1 3 x 7 5 9 x 7 25 16 x 32
20. Step 1: Step 2:
21. Step 1: Step 2:
or 14 0.36 0.11; thus 0.09.
19 x 3 1 1 19 x 3 1 2 19 x 4 4 19 x 16
19. Step 1: Step 2:
9 25
x 23 x 23 23 x 23. Then 23 16 7, or 23 32 9; thus 7. 1 x
0.36.
1 0.00248, or 1 1 1 1 0.00251; 12 2.01 2 1.99 1.99 2
15. Step 1: Step 2:
Step 2:
0.1340;
14 0.05 0.05 1x 14 0.05 0.2
1 x
0.3 10 x 10 or 10 x 5. 2 3 3
x 4 x 4 4 x 4. Then 4 10 or 23 , or 4 5 or 1; thus 23 . 3
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78
Chapter 2 Limits and Continuity
22. Step 1: Step 2:
x 2 3 0.1 0.1 x 2 3 0.1 2.9 x 2 3.1 2.9 x 3.1 x 3 x 3 3 x 3.
Then 3 2.9 3 2.9 0.0291, or 3 3.1 3.1 3 0.0286; thus 0.0286 23. Step 1: Step 2:
24. Step 1: Step 2:
25. Step 1: Step 2:
26. Step 1: Step 2:
x 2 4 0.5 0.5 x 2 4 0.5 3.5 x 2 4.5 3.5 x 4.5 4.5 x 3.5,
for x near 2. x (2) x 2 2 x 2. Then 2 4.5 4.5 2 0.1213, or 2 3.5 2 3.5 0.1292; thus 4.5 2 0.12. 1 x
11 (1) 0.1 0.1 1x 1 0.1 10
1 x
9 10 x 10 or 10 x 10 . 10 11 9 9 11
x (1) x 1 1 x 1. 1 ; thus 1 . Then 1 10 19 , or 1 10 11 9 11 11
( x 2 5) 11 1 x 2 16 1 1 x 2 16 1 15 x 2 17 15 x 17.
x 4 x 4 4 x 4. Then 4 15 4 15 0.1270, or 4 17 17 4 0.1231; thus 17 4 0.12. 120 x
5 1 1 120 5 1 4 120 6 x x
1 4
x 120
1 6
30 x 20 or 20 x 30.
x 24 x 24 24 x 24. Then 24 20 4, or 24 30 6; thus 4.
27. Step 1: Step 2:
mx 2m 0.03 0.03 mx 2m 0.03 0.03 2m mx 0.03 2m 2 0.03 x 2 m x 2 x 2 2 x 2. Then 2 2 0.03 0.03 , or 2 2 0.03 0.03 . In either case, 0.03 . m m m m m
28. Step 1: Step 2:
mx 3m c c mx 3m c c 3m mx c 3m 3 mc x 3 mc x 3 x 3 3 x 3. Then 3 3 mc mc , or 3 3 mc mc . In either case, mc .
29. Step 1:
(mx b)
Step 2:
m2 b c c mx m2 c c m2 mx c m2 12 mc x 12 mc .
x 12 x 12 12 x 12 .
Then 12 12 mc 30. Step 1: Step 2:
c, m
or 12
1 2
mc
c . In m
either case,
c. m
(mx b) (m b) 0.05 0.05 mx m 0.05 0.05 m mx 0.05 m 1 0.05 x 1 0.05 . m m
x 1 x 1 1 x 1. Then 1 1 0.05 0.05 , or 1 1 0.05 m m m
0.05 . In m
either case,
0.05 . m
31. lim (3 2 x) 3 2(3) 3 x 3
Step 1:
(3 2 x ) (3) 0.02 0.02 6 2 x 0.02 6.02 2 x 5.98 3.01 x 2.99 or 2.99 x 3.01.
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0.03 . m
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Section 2.3 The Precise Definition of a Limit 0 x 3 x 3 3 x 3. Then 3 2.99 0.01, or 3 3.01 0.01; thus 0.01.
Step 2:
32.
lim (3 x 2) (3)(1) 2 1
x 1
Step 1: Step 2:
(3 x 2) 1 0.03 0.03 3 x 3 0.03 0.01 x 1 0.01 1.01 x 0.99. x (1) x 1 1 x 1. Then 1 1.01 0.01, or 1 0.99 0.01; thus 0.01.
x2 4 x 2 x 2
lim
33. lim
x 2 x2 4 x 2
lim
x 2 6x 5 x 5
Step 1:
4 0.05 3.95 x 2 4.05, x 2
2
lim
x5
x 6 x 5 x 5
(x 5)(x 1) (x 5)
lim ( x 1) 4, x 5. x 5
(4) 0.05 0.05
( x 5)( x 1) ( x 5)
4 0.05 4.05 x 1 3.95, x 5
1 5 x 4 0.5 0.5 1 5 x 4 0.5 3.5 1 5 x 4.5 12.25 1 5 x 20.25
Step 1: Step 2:
4 x2 x
( x 2)( x 2) ( x 2)
1 5 x 1 5(3) 16 4
lim
x 3
36. lim
x 2
5.05 x 4.95, x 5. x (5) x 5 5 x 5. Then 5 5.05 0.05, or 5 4.95 0.05; thus 0.05.
Step 2:
35.
lim ( x 2) 2 2 4, x 2
1.95 x 2.05, x 2. x 2 x 2 2 x 2. Then 2 1.95 0.05, or 2 2.05 0.05; thus 0.05.
Step 2:
x 5
( x 2)( x 2) ( x 2)
4 0.05 0.05
Step 1:
34.
79
Step 1: Step 2:
11.25 5 x 19.25 3.85 x 2.25. x (3) x 3 3 x 3. Then 3 3.85 0.85, or 3 2.25 0.75; thus 0.75. 4 2
2 4 x
2 0.4 0.4 4x 2 0.4 1.6
4 x
2.4 10 16
x 4
10 10 x 10 or 53 x 52 . 24 4 6
x 2 x 2 2 x 2. Then 2 53 13 , or 2 52 12 ; thus 13 .
37. Step 1: Step 2:
(9 x) 5 4 x 4 x 4 4 x 4 4 x 4 . x 4 x 4 4 x 4. Then 4 4 , or 4 4 . Thus choose .
38. Step 1: Step 2:
(3x 7) 2 3x 9 9 3 x 9 3 3 x 3 3 . x 3 x 3 3 x 3. Then 3 3 3 3 , or 3 3 3 3 . Thus choose 3 .
39. Step 1: Step 2:
x 5 2 x 5 2 2 x 5 2 (2 ) 2 x 5 (2 )2 (2 ) 2 5 x (2 ) 2 5. x 9 x 9 9 x 9.
Then 9 2 4 9 4 2 , or 9 2 4 9 4 2 . Thus choose the smaller distance, 4 2 . Copyright 2018 Pearson Education, Inc.
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80
Chapter 2 Limits and Continuity 4 x 2 4 x 2 2 4 x 2 (2 ) 2 4 x (2 ) 2
40. Step 1: Step 2:
41. Step 1: Step 2:
(2 ) 2 x 4 (2 )2 (2 )2 4 x (2 )2 4. x 0 x . Then (2 ) 2 4 2 4 4 2 , or (2 ) 2 4 4 2 . Thus choose the smaller distance, 4 2 .
For x 1, x 2 1 x 2 1 1 x 2 1 1 x 1 1 x 1 near x 1. x 1 x 1 1 x 1. Then 1 1 1 1 , or 1 1 1 1. Choose
min 1 1 , 1 1 , that is, the smaller of the two distances. 42. Step 1: Step 2:
For x 2, x 2 4 x 2 4 4 x 2 4 4 x 4 4 x 4 near x 2. x (2) x 2 2 x 2. Then 2 4 4 2, or 2 4 2 4 . Choose min 4 2, 2 4 .
43. Step 1: Step 2:
44. Step 1:
1 x
1 1x 1 1 1x 1 11 x 11 .
x 1 x 1 1 x 1 . Then 1 11 1 11 1 , or 1 11 11 1 1 . Choose 1 , the smaller of the two distances. 1 x2
13 12 13 13 x
133 x 2 133
Step 2:
Choose min
Step 2:
46. Step 1: Step 2:
47. Step 1: Step 2:
x
13 133 12 133 x
3 , 13
or
3 13
x
3 13
for x near 3.
x 3 x 3 3 x 3 .
Then 3
45. Step 1:
3 13
1 x2
3 13
3 133 , or 3 133 133 3. 3 133 , 133 3 .
(6) ( x 3) 6 , x 3 x 3 3 x 3. x 2 9 x 3
x (3) x 3 3 x 3. Then 3 3 , or 3 3 . Choose .
2 (x 1) 2 , x 1 1 x 1 . x 2 1 x 1
x 1 x 1 1 x 1 . Then 1 1 , or 1 1 . Choose . x 1: (4 2 x) 2 0 2 2 x since x 1. Thus, 1 2 x 0; x 1: (6 x 4) 2 0 6 x 6 since x 1. Thus, 1 x 1 6 . x 1 x 1 1 x 1 . Then 1 1 2 2 , or 1 1 6 6 . Choose 6 .
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Section 2.3 The Precise Definition of a Limit 48. Step 1: Step 2:
81
x 0: 2 x 0 2 x 0 2 x 0;
x 0: 2x 0 0 x 2. x 0 x . Then 2 2 , or 2 2. Choose 2 .
49. By the figure, x x sin 1x x for all x 0 and x x sin 1x x for x 0. Since lim ( x) lim x 0, then by x 0
the sandwich theorem, in either case, lim x sin 1x 0.
x 0
x 0
50. By the figure, x 2 x 2 sin 1x x 2 for all x except possibly at x 0. Since lim ( x 2 ) lim x 2 0, then by the x 0
sandwich theorem, lim x 2 sin 1x 0.
x 0
x 0
51. As x approaches the value 0, the values of g ( x) approach k. Thus for every number 0, there exists a 0 such that 0 x 0 g ( x) k . 52. Write x h c. Then 0 x c x c , x c ( h c) c , h c c h , h 0 0 h 0 . Thus, lim f ( x) L for any 0, there exists 0 such that f ( x) L whenever 0 x c x c
f (h c) L whenever 0 h 0 lim f (h c) L. h 0
53. Let f ( x) x 2 . The function values do get closer to 1 as x approaches 0, but lim f ( x) 0, not 1. The x 0
function f ( x) x 2 never gets arbitrarily close to 1 for x near 0.
54. Let f ( x) sin x, L 12 , and x0 0. There exists a value of x (namely x 6 ) for which sin x 12 for any
given 0. However, lim sin x 0, not 12 . The wrong statement does not require x to be arbitrarily close to x0 . x 0
As another example, let g ( x) sin 1x , L 12 , and x0 0. We can choose infinitely many values of x near 0 such that sin 1x 12 as you can see from the accompanying figure. However, lim sin 1x fails to exist. The wrong x 0
statement does not require all values of x arbitrarily close to x0 0 to lie within 0 of L 12 . Again you can see from the figure that there are also infinitely many values of x near 0 such that sin 1x 0. If we choose 14
we cannot satisfy the inequality sin 1x 12 for all values of x sufficiently near x0 0.
55.
2 π x2 A 9 0.01 0.01 2x 9 0.01 8.99 4 9.01 π4 (8.99) x 2 π4 (9.01)
2 8.99 x 2 9.01 or 3.384 x 3.387. To be safe, the left endpoint was rounded up and
the right endpoint was rounded down.
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82
Chapter 2 Limits and Continuity
56. V RI VR I
(120)(10) 51
R
V 5 R (120)(10) 49
0.1 0.1 120 5 0.1 4.9 120 5.1 10 R 10 R R 49 120 51 23.53 R 24.48.
To be safe, the left endpoint was rounded up and the right endpoint was rounded down. 57. (a) x 1 0 1 x 1 f ( x) x. Then f ( x) 2 x 2 2 x 2 1 1. That is, f ( x) 2 1
1 2
no matter how small is taken when 1 x 1 lim f ( x) 2. x 1
(b) 0 x 1 1 x 1 f ( x) x 1. Then f ( x) 1 ( x 1) 1 x x 1. That is, f ( x) 1 1 no matter how small is taken when 1 x 1 lim f ( x) 1. x 1
(c) x 1 0 1 x 1 f ( x) x. Then f ( x ) 1.5 x 1.5 1.5 x 1.5 1 0.5. Also, 0 x 1 1 x 1 f ( x) x 1. Then f ( x) 1.5 ( x 1) 1.5 x 0.5 x 0.5 1 0.5 0.5. Thus, no matter how small is taken, there exists a value of x such that x 1 but f ( x ) 1.5 12 lim f ( x) 1.5. x 1
58. (a) For 2 x 2 h( x) 2 h( x) 4 2. Thus for 2, h( x) 4 whenever 2 x 2 no matter how small we choose 0 lim h( x) 4. x2
(b) For 2 x 2 h( x ) 2 h( x) 3 1. Thus for 1, h( x) 3 whenever 2 x 2 no matter how small we choose 0 lim h( x) 3. x2
(c) For 2 x 2 h( x) x 2 so h( x) 2 x 2 2 . No matter how small 0 is chosen, x 2 is close to 4 when x is near 2 and to the left on the real line x 2 2 will be close to 2. Thus if 1, h( x) 2 whenever 2 x 2 no matter how small we choose 0 lim h( x) 2. x2
59. (a) For 3 x 3 f ( x) 4.8 f ( x) 4 0.8. Thus for 0.8, f ( x) 4 whenever 3 x 3 no matter how small we choose 0 lim f ( x) 4. x 3
(b) For 3 x 3 f ( x) 3 f ( x) 4.8 1.8. Thus for 1.8, f ( x) 4.8 whenever 3 x 3 no matter how small we choose 0 lim f ( x) 4.8. x 3
(c) For 3 x 3 f ( x) 4.8 f ( x) 3 1.8. Again, for 1.8, f ( x) 3 whenever 3 x 3 no matter how small we choose 0 lim f ( x) 3. x 3
60. (a) No matter how small we choose 0, for x near 1 satisfying 1 x 1 , the values of g ( x) are near 1 g ( x) 2 is near 1. Then, for 12 we have g ( x) 2 12 for some x satisfying 1 x 1 , or 0 x 1 lim g ( x) 2. x 1
(b) Yes, lim g ( x ) 1 because from the graph we can find a 0 such that g ( x) 1 if 0 x (1) . x 1
61–66. Example CAS commands (values of del may vary for a specified eps): Maple: f : x - (x^4-81)/(x-3); x0 : 3;
.
plot( f (x), x x0-1..x0 1, color black, title "Section 2.3, #61(a)" );
# (a)
L : limit( f (x), x x0 );
# (b)
epsilon : 0.2; # (c) plot( [f (x), L-epsilon,L epsilon], x x0-0.01..x0 0.01, color black, linestyle [1,3,3], title "Section 2.3, #61(c)" );
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Section 2.4 One-Sided Limits q : fsolve( abs( f (x)-L ) epsilon, x x0-1..x0 1 ); # (d) delta : abs(x0-q); plot( [f (x), L-epsilon, L epsilon], x x0-delta..x0 delta, color black, title "Section 2.3, #61(d)" ); for eps in [0.1, 0.005, 0.001 ] do # (e) q : fsolve( abs( f (x)-L ) eps, x x0-1..x0 1 ); delta : abs(x0-q); head : sprintf ("Section 2.3, #61(e)\n epsilon %5f , delta %5f \n", eps, delta ); print(plot( [f (x), L-eps, L eps], x x0-delta..x0 delta, color black, linestyle [1,3,3], title head ));
end do: Mathematica (assigned function and values for x0, eps and del may vary): Clear[f , x] y1: L eps; y2: L eps; x0 1; f[x _]: (3x 2 (7x 1)Sqrt[x] 5)/(x 1) Plot[f [x], {x, x0 0.2, x0 0.2}] L: Limit[f [x], x x0] eps 0.1; del 0.2; Plot[{f [x], y1, y2}, {x, x0 del, x0 del}, PlotRange {L 2eps, L 2eps}]
2.4
ONE-SIDED LIMITS
1. (a) True (e) True (i) False
(b) True (f) True (j) False
(c) False (g) False (k) True
(d) True (h) False (l) False
2. (a) True (e) True (i) True
(b) False (f) True (j) False
(c) False (g) True (k) True
(d) True (h) True
3. (a)
lim f ( x) 22 1 2, lim f ( x) 3 2 1 x 2
x 2
(b) No, lim f ( x) does not exist because lim f ( x) lim f ( x) x 2
x 2
(c)
x 2
lim f ( x ) 42 1 3, lim f ( x) 42 1 3
x 4
x 4
(d) Yes, lim f ( x) 3 because 3 lim f ( x) lim f ( x) x 4
x 4
4. (a)
x 4
lim f ( x) 22 1, lim f ( x) 3 2 1, f (2) 2 x 2
x 2
(b) Yes, lim f ( x) 1 because 1 lim f ( x) lim f ( x ) x 2
x 2
(c)
lim
x 1
f ( x) 3 (1) 4, lim
x 1
f ( x) 3 (1) 4
(d) Yes, lim f ( x ) 4 because 4 lim
x 1
x 1
5. (a) No, lim f ( x) does not exist since sin (b)
x 0
x 2
f ( x) lim
x 1
f ( x)
1x does not approach any single value as x approaches 0
lim f ( x) lim 0 0
x 0
x 0
(c) lim f ( x) does not exist because lim f ( x) does not exist x 0
x 0
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84
Chapter 2 Limits and Continuity
6. (a) Yes, lim g ( x) 0 by the sandwich theorem since x g ( x) x when x 0 x 0
(b) No, lim g ( x) does not exist since x is not defined for x 0 x 0
(c) Yes, lim g ( x) lim g ( x) 0 since x 0 is a boundary point of the domain x 0
x 0
7. (a)
(b)
lim f ( x) 1 lim f ( x )
x 1
x 1
(c) Yes, lim f ( x) 1 since the right-hand and left-hand x 1
limits exist and equal 1
8. (a)
(b)
lim f ( x) 0 lim f ( x )
x 1
x 1
(c) Yes, lim f ( x) 0 since the right-hand and left-hand x 1
limits exist and equal 0
9. (a) domain: 0 x 2 range: 0 y 1 and y 2 (b) lim f ( x) exists for c belonging to (0, 1) (1, 2) x c
(c) x 2 (d) x 0
10. (a) domain: x range: 1 y 1 (b) lim f ( x) exists for c belonging to x c
( , 1) ( 1, 1) (1, ) (c) none (d) none
11.
13.
lim
x 0.5
lim
x 2
x2 x 1
0.5 2 0.51
3/2 1/2
3
12.
lim
x 1
x 1 x2
1112 0 0
xx1 2x x5x 221 (2(2)2)(52) (2) 12 1 2
2
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Section 2.4 One-Sided Limits 14.
15.
x11 xx 6 37 x 111 116 371 12 17 72 1
lim
x 1
lim
h 0
h2 4h 5 5 h
2 2 ( h 2 4 h 5) 5 lim h 4 hh5 5 h 2 4h 5 5 lim h 0 h 4 h 5 5 h 0 h h 2 4 h 5 5
lim
h 0 h
16.
lim
h 0
6 5h 2 11h 6 h
h ( h 4)
h2 4h 5 5
0 4 5 5
2 5
2 2 lim 6 5hh 11h 6 6 5h2 11h 6 h 0 6 5h 11h 6
lim
h 0 h
6 (5h 2 11h 6)
6 5h 2 11h 6
x2
lim
h 0 h
h (5h 11) 6 5h 2 11h 6
(0 11) 6 6
11
2 6
( x 2)
lim ( x 3) x 2 lim ( x 3) ( x 2) (|x 2| ( x 2) for x 2) x 2 x 2 lim ( x 3) ((2) 3) 1
17. (a)
lim ( x 3)
(b)
x 2
x2 x2
x 2
( x 2) lim ( x 3) ( x 2) (|x 2| ( x 2) for x 2) x 2 lim ( x 3)(1) (2 3) 1 x 2
18. (a)
2 x ( x 1) x 1
lim
x 1
(b)
lim
2 x ( x 1) ( x 1)
x 1
lim 2 x ( x 1) x 1
lim
x 1
x 1
lim
x 1
(|x 1| x 1 for x 1)
2x 2 2 x ( x 1) ( x 1)
(|x 1| ( x 1) for x 1)
lim 2 x 2 x 1
19. (a) If 0 x 2 , then sin x 0, so that lim x 0
sin x sin x
lim
x 0
sin x sin x
lim 1 1 x 0
sin x sin x lim 1 1 (b) If 2 x 0, then sin x 0, so that lim sin x lim sin x x 0 x 0 x 0 cos x lim 1cos x lim 1 1 20. (a) If 0 x 2 , then cos x 1, so that lim 1cos x lim 1(cos 1cos x x 1) x 0 cos x 1 x 0 x 0 x 0 cos x 1 lim 1 1 (b) If 2 x 0, then cos x 1, so that lim cos x 1 lim (cos x 1) x 0 cos x 1
lim 33 1 3
21. (a)
lim (t t ) 4 4 0
22. (a)
23.
lim sin 2 lim sinx x 1
t 0
(b)
t 4
0
24. lim
(b)
2
sin kt t
x 0
x 0
x 0
lim 23 3
lim (t t ) 4 3 1
t 4
(where x 2 )
kt lim k sin k lim sin k 1 k lim k sin 0 0 t 0 kt
(where kt )
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86
Chapter 2 Limits and Continuity sin 3 y y 0 4 y
26.
lim
h 0
h sin 3h
tan 2 x x 0 x
27. lim
2t t 0 tan t
28. lim
1 lim 3sin 3 y 4 y 0 3 y
25. lim
h 0
x csc 2 x x 0 cos 5 x
29. lim
sin 2 x x 0 x cos 2 x
x
x 0
t
sin t cos t
t cos t t 0 sin t
2 lim
x 1 x 0 sin 2 x cos 5 x
lim
sin3h3h
x x cos x x 0 sin x cos x
x 2 x sin x 2x x 0
x x 0 2
lim
12 12
(where 3y ) 1 1 1 3 3
(where 3h)
lim 2sin 2 x 1 2 2 x 0 2 x
1sin t lim t t 0
2 1 1 2
lim 3cos x sinx x sin2 x2 x 3 1 1 3 x 0
x sinx cos lim x cos x
lim sin1 x lim x 0 x x 0
32. lim
3 4
2 x lim 1 1 1 (1) 1 12 xlim 2 2 0 sin 2 x x 0 cos 5 x
x x 0 sin x cos x
lim
1 cos 2 x
2 lim cos t t 0
6 x 2 cos x x sin 2 x sin x 0
31. lim
1 13 lim sin 0
lim x 0
30. lim 6 x 2 (cot x)(csc 2 x ) lim x 0
3 lim sin 4 0
1
sin 2 x cos 2x lim
lim
t 0
3 lim sin 3 y 4 y 0 3 y
13 sin3h3h 13 hlim 0
lim
2 lim
x 1 x 0 sin x cos x
cos1 x xlim 0
1
sin x x
x xlim 0 sin x
(1)(1) 1 2
sinx x 0 12 12 (1) 0
1cos lim (1cos )(1 cos ) lim 1cos 2 sin 2 lim 0 sin 2 0 (2sin cos )(1 cos ) 0 (2sin cos )(1 cos ) 0 (2sin cos )(1 cos )
33. lim
sin 0 0 0 (2 cos )(1 cos ) (2)(2)
lim
x x cos x 2 x 0 sin 3 x
34. lim
lim
x 0
sin(1cos t ) t 0 1cos t
35. lim
36. lim
h 0
sin(sin h ) sin h
x (1cos x ) 2
sin 3 x
lim
x 0
x (1 cos x ) 9 x2 sin 2 3 x
lim
9 x2
x 0
1 cos x 9x
sin3 x3 x
2
1 lim 1 cos x 9 x0 x 2 sin 3 x lim 3 x x 0
1 (0) 9 2
1
0
lim sin 1 since 1 cos t 0 as t 0 0
lim sin 1 since sin h 0 as h 0 0
sin lim sin 2 1 lim sin 2 12 1 1 12 2 0 sin 2 0 sin 2 0 sin 2 2
37. lim
sin 5 x x 0 sin 4 x
38. lim
sin 5 x 4 x 5 x 0 sin 4 x 5 x 4
lim
54 xlim sin5 x5x sin4 x4 x 54 11 54 0
39. lim cos 0 1 0 0
40. lim sin cot 2 lim sin 0
0
cos 2 sin 2
lim sin 0
cos 2 2sin cos
cos 2 1 2 0 2 cos
lim
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Section 2.4 One-Sided Limits tan 3 x x 0 sin 8 x
41. lim
87
xlim sin 3x 1 8 x 3 0 cos 3 x sin 8 x 3 x 8 83 lim cos13 x sin3 x3 x sin8 x8 x 83 1 1 1 83 x0 sin 3 x 1 x 0 cos 3 x sin 8 x
lim
sin 3 y cot 5 y y 0 y cot 4 y
lim
y 0
tan 2 0 cot 3
43. lim
1 1 1 1
sin 3 y sin 4 y cos 5 y y 0 y cos 4 y sin 5 y
lim
42. lim
lim
0
sin 3 y 3y
sin cos 3 2 cos sin 3
sin 4 y 4y
sin 3 y y y 0
sin 4 y cos 4 y
lim
5y sin 5 y
cos 5 y cos 4 y
34 5
345 y 345 y
12 5
12 5
sin33 cos 3cos 3 (1)(1) 113 3
sin sin 3 lim sin 2 0 cos cos 3 0
lim
cos 5 y sin 5 y
4 2 2 2 cos cot 4 lim cos 4 sin 2 2 lim cos 4 (2sin cos ) lim cos 4 (4sin cos ) sin 4 lim 2 2 2 2 2 2 2 2 2 0 sin cos 2 sin 4 0 sin cot 2 0 sin 2 cos 2 0 sin cos 2 sin 4 0 sin cos 2 sin 4 2
44. lim
sin 2
111 1
1 cos 4 cos2 1 cos 4 cos2 4 lim lim 1 2 2 sin 4 2 0 cos 2 sin 4 0 sin 4 cos 2 0 4 cos 2
lim
45.
lim
x 0
1cos3 x 2x
lim
x 0
lim
1cos3 x 1 cos3 x 2x 1 cos3 x 3 sin
0 2
46.
cos 2 x cos x x2 x 0
lim
lim
4 cos 4 cos 2
x 0
1cos 2 3 x x 0 2 x (1 cos3 x )
lim
3 (1) 1sin cos 2
cos x (cos x 1) x2
101 0
sin 2 3 x x 0 2 x (1 cos3 x )
lim
cos x (cos x 1) cos x 1 cos x 1 x2 x 0
lim
3 sin 3 x sin 3 x x 0 2 3 x 1 cos3 x
lim
(where 3x )
cos x (1)(1) 1 lim sinx x sinx x cos x 1 11 x 0
2
2
lim
x 0
cos x (cos 2 x 1) x 2 (cos x 1)
lim
x 0
cos x( sin 2 x ) x 2 (cos x 1)
1 2
47. Yes. If lim f ( x) L lim f ( x), then lim f ( x) L. If lim f ( x ) lim f ( x), then lim f ( x) does not exist.
x a
xa
x a
x a
x a
x a
48. Since lim f ( x) L if and only if lim f ( x) L and lim f ( x) L, then lim f ( x) can be found by x c
calculating lim f ( x).
x c
x c
x c
x c
49. If f is an odd function of x, then f ( x) f ( x). Given lim f ( x) 3, then lim f ( x) 3. x 0
x 0
50. If f is an even function of x, then f ( x) f ( x). Given lim f ( x) 7 then lim can be said about lim
x 2
x 2
f ( x ) because we don’t know lim f ( x).
x 2
f ( x) 7. However, nothing
x 2
51. I (5, 5 ) 5 x 5 . Also, x 5 x 5 2 x 5 2 . Choose 2 lim
x 5 0.
52. I (4 , 4) 4 x 4. Also, 4 x 4 x 2 x 4 2 . Choose 2 lim
4 x 0.
x 5
x 4
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88
Chapter 2 Limits and Continuity
53. As x 0 the number x is always negative. Thus,
x x
(1)
x x
1 0 which is always true x x 0 x
independent of the value of x. Hence we can choose any 0 with x 0 lim 54. Since x 2 we have x 2 and x 2 x 2. Then,
x2 x2
1
x2 x2
1 0 which is always true so
long as x 2. Hence we can choose any 0, and thus 2 x 2 55. (a)
lim
x 400
1.
x2 x2
x2 x 2 x 2
1 . Thus, lim
1.
x 400. Just observe that if 400 x 401, then x 400. Thus if we choose 1, we have for
any number 0 that 400 x 400 x 400 400 400 0 . (b) lim x 399. Just observe that if 399 x 400 then x 399. Thus if we choose 1, we have for x 400
any number 0 that 400 x 400 x 399 399 399 0 . (c) Since lim x lim x we conclude that lim x does not exist. x 400
56. (a)
x 400
lim f ( x) lim
x 0
x 0
x 400
x 0 0;
x 0 x 0 x 2 for x positive. Choose 2
lim f ( x) 0. x 0
(b)
1x 0 by the sandwich theorem since x2 x2 sin 1x x2 for all x 0. Since x 2 0 x 2 0 x 2 whenever x , we choose and obtain x 2 sin 1x 0 lim f ( x) lim x 2 sin
x 0
x 0
if x 0. (c) The function f has limit 0 at x0 0 since both the right-hand and left-hand limits exist and equal 0. 2.5
CONTINUITY
1. No, discontinuous at x 2, not defined at x 2 2. No, discontinuous at x 3, 1 lim g ( x) g (3) 1.5 x 3
3. Continuous on [1, 3] 4. No, discontinuous at x 1, 1.5 lim k ( x) lim k ( x) 0 x 1
x 1
5. (a) Yes
(b) Yes, lim
(c) Yes
(d) Yes
6. (a) Yes, f (1) 1
x 1
f ( x) 0
(b) Yes, lim f ( x) 2
(c) No
(d) No
7. (a) No
(b) No
x 1
8. [1, 0) (0, 1) (1, 2) (2, 3) 9. f (2) 0, since lim f ( x) 2(2) 4 0 lim f ( x) x 2
x 2
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Section 2.5 Continuity 10. f (1) should be changed to 2 lim f ( x ) x 1
11. Nonremovable discontinuity at x 1 because lim f ( x) fails to exist ( lim f ( x) 1 and lim f ( x) 0). x 1
x 1
x 1
Removable discontinuity at x 0 by assigning the number lim f ( x) 0 to be the value of f (0) rather x 0
than f (0) 1.
12. Nonremovable discontinuity at x 1 because lim f ( x) fails to exist ( lim f ( x) 2 and lim f ( x ) 1). x 1
x 1
x 1
Removable discontinuity at x 2 by assigning the number lim f ( x) 1 to be the value of f (2) rather than x 2
f (2) 2.
14. Discontinuous only when ( x 2)2 0 x 2
13. Discontinuous only when x 2 0 x 2
15. Discontinuous only when x 2 4 x 3 0 ( x 3)( x 1) 0 x 3 or x 1 16. Discontinuous only when x 2 3 x 10 0 ( x 5)( x 2) 0 x 5 or x 2 17. Continuous everywhere. (|x 1| sin x defined for all x; limits exist and are equal to function values.) 18. Continuous everywhere. (|x| 1 0 for all x; limits exist and are equal to function values.) 19. Discontinuous only at x 0 20. Discontinuous at odd integer multiples of 2 , i.e., x (2n 1) 2 , n an integer, but continuous at all other x. 21. Discontinuous when 2x is an integer multiple of , i.e., 2 x n , n an integer x n2 , n an integer, but continuous at all other x. 22. Discontinuous when 2x is an odd integer multiple of 2 , i.e., 2x (2n 1) 2 , n an integer x 2n 1, n an integer (i.e., x is an odd integer). Continuous everywhere else. 23. Discontinuous at odd integer multiples of 2 , i.e., x (2n 1) 2 , n an integer, but continuous at all other x. 24. Continuous everywhere since 1 sin x 1 0 sin 2 x 1 1 sin 2 x 1; limits exist and are equal to the function values.
25. Discontinuous when 2 x 3 0 or x 32 continuous on the interval 32 , .
26. Discontinuous when 3 x 1 0 or x 13 continuous on the interval 13 , . 27. Continuous everywhere: (2 x 1)1/3 is defined for all x; limits exist and are equal to function values. 28. Continuous everywhere: (2 x)1/5 is defined for all x; limits exist and are equal to function values. x 2 x 6 x 3 x 3
29. Continuous everywhere since lim
( x 3)( x 2) x 3 x 3
lim
lim ( x 2) 5 g (3) x 3
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90
Chapter 2 Limits and Continuity
30. Discontinuous at x 2 since lim f ( x) does not exist while f (2) 4. x 2
31. Discontinuous at x 1; lim ( x 2 2) 3, but lim e x e, so that lim f ( x) does not exist while f (1) e; x 1
x 1
x 1
x
and lim (1 x) 1 lim e , so that lim f ( x) 1 f (0) x 0
x 0
x 0
32. Discontinuous at x ln 2, since 2 e x 0 e x 2 ln e x ln 2 x ln 2 33. lim sin( x sin x) sin( sin ) sin( 0) sin 0, and function continuous at x . x
34. lim sin( 2 cos(tan t )) sin( 2 cos(tan(0))) sin 2 cos(0) sin 2 1, and function continuous at t 0. t 0
35. lim sec ( y sec2 y tan 2 y 1) lim sec ( y sec2 y sec2 y ) lim sec (( y 1) sec2 y ) sec ((1 1)sec2 1) y 1
y 1
y 1
sec 0 1, and function continuous at y 1.
36. lim tan 4 cos(sin x1/3 ) tan 4 cos(sin(0)) tan 4 cos(0) tan 4 1, and function continuous at x 0. x 0 37. lim cos t 0
19 3 sec 2t
cos
193 sec 0
cos cos 4 16
2 2
, and function continuous at t 0.
csc2 x 5 3 tan x csc2 6 5 3 tan 6 4 5 3
38. lim
x 6
at x 6 . 39.
40.
lim sin 2 e
x 0
x
sin e sin 0 2
1 3
9 3, and function continuous
1, and the function is continuous at x = 0. 2
lim cos 1 ln x cos 1 ln 1 cos 1 (0) 2 , and the function is continuous at x = 1.
x 1
41. g ( x) 42. h(t )
x 2 9 x 3
( x 3)( x 3) ( x 3)
t 2 3t 10 t 2
(t 5)(t 2) t 2
( s 2 s 1)( s 1) ( s 1)( s 1)
43. f ( s )
s 3 1 s 3 1
44. g ( x)
x 2 16 x 2 3 x 4
x 3, x 3 g (3) lim ( x 3) 6
x 3
t 5, t 2 h(2) lim (t 5) 7 t 2
( x 4)( x 4) ( x 4)( x 1)
s 2 s 1 , s 1
x 4 , x 1
2 s 1 f (1) lim s ss11 s 1
x 4 g (4) lim
x 4
3 2
xx14 85
45. As defined, lim f ( x) (3)2 1 8 and lim (2a )(3) 6a. For f ( x) to be continuous we must have x 3
6a 8 a 43 .
x 3
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Section 2.5 Continuity
91
46. As defined, lim g ( x) 2 and lim g ( x) b(2)2 4b. For g ( x) to be continuous we must have x 2
x 2
4b 2 b 12 .
47. As defined, lim f ( x) 12 and lim f ( x) a 2 (2) 2a 2a 2 2a. For f ( x ) to be continuous we must have 2
x 2
x 2
12 2a 2a a 3 or a 2.
48. As defined, lim g ( x) b b 1
x 0
0 b b 1
b b 1
and lim g ( x) (0) 2 b b. For g ( x) to be continuous we must have
b b 0 or b 2.
49. As defined, lim
x 1
f ( x) 2 and lim
x 0
x 1
f ( x) a (1) b a b, and lim f ( x) a (1) b a b and x 1
lim f ( x) 3. For f ( x) to be continuous we must have 2 a b and a b 3 a
x 1
5 2
and b 12 .
50. As defined, lim g ( x) a (0) 2b 2b and lim g ( x) (0)2 3a b 3a b, and lim g ( x) (2)2 3a b x 0
x 0
x 2
4 3a b and lim g ( x) 3(2) 5 1. For g ( x) to be continuous we must have 2b 3a b and 4 3a b 1 x 0
a 32 and b 32 .
51. The function can be extended: f (0) 2.3.
52. The function cannot be extended to be continuous at x 0. If f (0) 2.3, it will be continuous from the right. Or if f (0) 2.3, it will be continuous from the left.
53. The function cannot be extended to be continuous at x 0. If f (0) 1, it will be continuous from the right. Or if f (0) 1, it will be continuous from the left.
54. The function can be extended: f (0) 7.39.
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92
Chapter 2 Limits and Continuity
55. f ( x) is continuous on [0, 1] and f (0) 0, f (1) 0 by the Intermediate Value Theorem f ( x) takes on every value between f (0) and f (1) the equation f ( x) 0 has at least one solution between x 0 and x 1.
56. cos x x (cos x) x 0. If x 2 , cos 2 2 0. If x 2 , cos 2 2 0. Thus cos x x 0 for some x between and according to the Intermediate Value Theorem, since the function cos x x is
continuous.
2
2
57. Let f ( x) x3 15 x 1, which is continuous on [4, 4]. Then f (4) 3, f (1) 15, f (1) 13, and f (4) 5. By the Intermediate Value Theorem, f ( x ) 0 for some x in each of the intervals 4 x 1, 1 x 1, and 1 x 4. That is, x3 15 x 1 0 has three solutions in [4, 4]. Since a polynomial of degree 3 can have at most 3 solutions, these are the only solutions. 58. Without loss of generality, assume that a b. Then F ( x) ( x a )2 ( x b)2 x is continuous for all values of x, so it is continuous on the interval [a, b]. Moreover F (a ) a and F (b) b. By the Intermediate Value Theorem, since a a 2 b b, there is a number c between a and b such that F ( x) a 2 b . 59. Answers may vary. Note that f is continuous for every value of x. (a) f (0) 10, f (1) 13 8(1) 10 3. Since 3 10, by the Intermediate Value Theorem, there exists a c so that 0 c 1 and f (c) . (b) f (0) 10, f (4) (4)3 8(4) 10 22. Since 22 3 10, by the Intermediate Value Theorem, there exists a c so that 4 c 0 and f (c) 3. (c) f (0) 10, f (1000) (1000)3 8(1000) 10 999,992, 010. Since 10 5, 000, 000 999,992, 010, by the Intermediate Value Theorem, there exists a c so that 0 c 1000 and f (c) 5, 000, 000. 60. All five statements ask for the same information because of the intermediate value property of continuous functions. (a) A root of f ( x ) x3 3 x 1 is a point c where f (c) 0. (b) The point where y x3 crosses y 3x 1 have the same y-coordinate, or y x3 3 x 1 f ( x) x3 3 x 1 0. (c) x3 3x 1 x3 3 x 1 0. The solutions to the equation are the roots of f ( x) x3 3 x 1. (d) The points where y x3 3 x crosses y 1 have common y-coordinates, or y x3 3 x 1 f ( x) x3 3 x 1 0. (e) The solutions of x3 3x 1 0 are those points where f ( x) x3 3 x 1 has value 0. sin( x 2)
61. Answers may vary. For example, f ( x) x 2 is discontinuous at x 2 because it is not defined there. However, the discontinuity can be removed because f has a limit (namely 1) as x 2. 62. Answers may vary. For example, g ( x)
1 x 1
has a discontinuity at x 1 because lim g ( x) does not exist.
lim g ( x) and lim g ( x) . x 1 x 1
x 1
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Section 2.5 Continuity
93
63. (a) Suppose x0 is rational f ( x0 ) 1. Choose 12 . For any 0 there is an irrational number x (actually infinitely many) in the interval ( x0 , x0 ) f ( x) 0. Then 0 |x x0 | but | f ( x) f ( x0 )| 1 12 , so lim f ( x) fails to exist f is discontinuous at x0 rational. x x0
On the other hand, x0 irrational f ( x0 ) 0 and there is a rational number x in ( x0 , x0 ) f ( x) 1. Again lim f ( x) fails to exist f is discontinuous at x0 irrational. That is, f is discontinuous at every point. x x0
(b) f is neither right-continuous nor left-continuous at any point x0 because in every interval ( x0 , x0 ) or ( x0 , x0 ) there exist both rational and irrational real numbers. Thus neither limits lim f ( x) and x x0
lim f ( x) exist by the same arguments used in part (a).
x x0
64. Yes. Both f ( x) x and g ( x) x 12 are continuous on [0, 1]. However g
12 0
f ( x) g ( x)
f ( x) g ( x)
is undefined at x
1 2
since
is discontinuous at x 12 .
65. No. For instance, if f ( x) 0, g ( x) x , then h( x) 0 x 0 is continuous at x 0 and g ( x) is not. 66. Let f ( x) x11 and g ( x) x 1. Both functions are continuous at x 0. The composition f g f ( g ( x)) 1 1 is discontinuous at x 0, since it is not defined there. Theorem 10 requires that f ( x ) be continuous ( x 1) 1 x at g (0), which is not the case here since g (0) 1 and f is undefined at 1.
67. Yes, because of the Intermediate Value Theorem. If f (a ) and f (b) did have different signs then f would have to equal zero at some point between a and b since f is continuous on [a, b]. 68. Let f ( x ) be the new position of point x and let d ( x) f ( x) x. The displacement function d is negative if x is the left-hand point of the rubber band and positive if x is the right-hand point of the rubber band. By the Intermediate Value Theorem, d ( x) 0 for some point in between. That is, f ( x) x for some point x, which is then in its original position. 69. If f (0) 0 or f (1) 1, we are done (i.e., c 0 or c 1 in those cases). Then let f (0) a 0 and f (1) b 1 because 0 f ( x) 1. Define g ( x) f ( x) x g is continuous on [0, 1]. Moreover, g (0) f (0) 0 a 0 and g (1) f (1) 1 b 1 0 by the Intermediate Value Theorem there is a number c in (0, 1) such that g (c) 0 f (c) c 0 or f (c) c. f (c )
70. Let 2 0. Since f is continuous at x c there is a 0 such that x c f ( x) f (c) f (c) f ( x) f (c) . If f (c) 0, then 12 f (c) 12 f (c) f ( x) 23 f (c) f ( x) 0 on the interval (c , c ). If f (c) 0, then 12 f (c)
3 2
f (c ) f ( x )
1 2
f (c) f ( x) 0 on the interval (c , c ).
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94
Chapter 2 Limits and Continuity
71. By Exercise 52 in Section 2.3, we have lim f ( x ) L lim f (c h ) L. x c
h 0
Thus, f ( x) is continuous at x c lim f ( x) f (c) lim f (c h) f (c). x c
h 0
72. By Exercise 71, it suffices to show that lim sin(c h) sin c and lim cos(c h) cos c. h 0
h 0
Now lim sin(c h) lim (sin c)(cos h) (cos c)(sin h) (sin c) lim cos h (cos c) lim sin h . h 0 h 0 h 0 h 0 By Example 11 Section 2.2, lim cos h 1 and lim sin h 0. So lim sin(c h) sin c and thus f ( x) sin x is h 0
h 0
continuous at x c. Similarly,
h 0
lim cos(c h) lim (cos c)(cos h) (sin c)(sin h) (cos c) lim cos h (sin c) lim sin h cos c. Thus, h 0 h 0 h 0 g ( x) cos x is continuous at x c.
h 0
73. x 1.8794, 1.5321, 0.3473
74. x 1.4516, 0.8547, 0.4030
75. x 1.7549
76. x 1.5596
77. x 3.5156
78. x 3.9058, 3.8392, 0.0667
79. x 0.7391
80. x 1.8955, 0, 1.8955
2.6
LIMITS INVOLVING INFINITY; ASYMPTOTES OF GRAPHS
1. (a) lim f ( x) 0 (c) (e)
x 2
lim
x 3
(b)
f ( x) 2
lim f ( x) 1
(f)
x 0
(h)
x 0
lim f ( x) 0
x
2. (a) lim f ( x) 2 (c) (e) (g) (i)
(b)
x 4
lim f ( x) 1
x 2
lim
x 3
x 3
lim f ( x)
x 0
lim f ( x) 1
x
lim f ( x ) 3
x 2
(d) lim f ( x) does not exist
f ( x)
(f)
lim f ( x)
(h)
lim f ( x)
(j)
x 3
x 0
(k) lim f ( x) 0
(l)
x
1 m/ n x x
Note: In these exercises we use the result lim
Theorem 8 and the power rule in Theorem 1: lim
x 2
lim
x 3
1
lim f ( x)
lim f ( x) does not exist
x 0
lim f ( x) 1
x m n
(b) 3
4. (a)
(b) (b)
0. This result follows immediately from
1 x x
lim
3. (a) 3
1 2
f ( x)
x 0
0 whenever
m/n x x
5. (a)
f ( x) 2
(d) lim f ( x) does not exist
(g) lim f ( x) does not exist (i)
lim
x 3
m/ n
lim 1x x
m/ n
0m / n 0.
1 2
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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs
6. (a)
1 8
(b)
7. (a) 53 8. (a)
(b) 53
3 4
9. 1x
(b)
sin 2 x x
10. 31
cos 3
1 x
lim
x
sin 2 x x
31 lim
2t sin t t t cos t
11. lim
lim
t
r sin r r 2 r 7 5sin r
12. lim
14. (a)
2 x3 7 3 2 x x x x 7
0 by the Sandwich Theorem
cos 3
0 by the Sandwich Theorem
010 1 10
r
2 x 3 x 5 x 7
lim
lim 10 1 r 200 2
sin r r 7 2 r 5 sinr r
1
2 3x
lim
7 x 5 x
2 5
(b)
2 73 x 7 1 1 x 1 x 2 3
lim
lim
(b) 2 (same process as part (a))
16. (a)
17. (a)
x 1 2 x x 3
lim
3x 7 2 x x 2
lim
lim
lim
1 1 x x2
x 1
3
9 x4 x 4 2 x 2 x 5 x x 6
19. (a)
10 x5 x 4 31 x6 x
20.
x 3 7 x 2 2 2 x x x 1
(a) lim
2
0
(b) 0 (same process as part (a))
lim
x
lim
x
9
7 9
1
x3 2 52 13 64 x x x
10 1 31 x x 2 x6
1
0
lim
21. (a)
3 x 7 5 x 2 1 3 x 6 x 7 x 3
lim
9 2
(b)
9 2
(same process as part (a))
(b) 0 (same process as part (a))
, since x n 0 and x 7 .
x 7 2 x 1 1 2 x 1 x x
x 3 7 x 2 2 2 x x x 1
(b) 7 (same process as part (a))
x2
x 7 2 x 1 1 2 1 x x x
(b)
lim
x
x2
lim
lim
2
(b) 0 (same process as part (a))
7 3 x 1 x
lim
(same process as part (a))
0
lim
18. (a)
lim
3
x
2 5
x2
3 7 x x2
x 1
7 x3 2 x x 3 x 6 x
lim
3 4
2 1 sin t t t 1 cost t
lim
13. (a)
15. (a)
1 8
, since x n 0 and x 7 .
3 x 4 5 x 1 x 3 2 3 x 67 x 3 x
, since x n 0 and 3x 4 .
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96
Chapter 2 Limits and Continuity (b)
22. (a) (b)
23.
7 2 4 1 3 lim 3 x 35 x 1 lim 3 x 5x2 x3 , since x n 0 and 3x 4 .
x 6 x 7 x 3
8 3 3 2 5 lim 5 x 2 x 5 9 lim 5 x 25 x 49 x , since x n 0, 5x 3 , and the denominator 4.
x 3 x 4 x
3
x 3 x 4 x
8 x 2 3 2 x2 x
lim
x
25.
1 x3 2 x x 7 x
26. lim
x
lim
x 1 x 4 2 3 x x x
lim
x2 1 1x
x
lim
3
lim
32.
x 5 x 3 2/3 x 2 x x 4
33.
lim
x
x
x 2 1 x 1
lim
x 3
x
2
1 1x 12 x lim x 8 32 x
1/3
18000
1/3
18
1/3
1 2
10000 0 0
28.
1 21 x /15 x 1 1 x 2 /15
1 7 x19/15 x8/5 3 1 1 3/5 11/10 x x
x 2
1 4 x1/3 x
x 2/3
x 2 1/ x 2 2
lim 2 x x 2 x
2 1 1/ 2 lim x x 2 1 x1/ 2
1
1
x
x 2 1/ x 2
( x 2 1)/ x 2 ( x 1)/ x
11/ x 2 (1 x 1/ x )
lim
2
1 0 (1 0)
1
2
2 ( x 1)/ x lim ( x 1)/( x ) lim ( 111/1/xx ) ( 1100) 1 x x
( x 3)/ x 2 2
52
lim
x ( x 1)/ x 2
x
1 2 x 2 x3
lim
5 3x
lim
x 1
0
1
lim
1 5 x x2
lim
2 x1/15
x ( x 1)/ x
4 x 25
42
x
lim
lim
1/3
8 0 2 0
1
x
2 x5/3 x1/3 7 8/5 x x 3 x x
x 2 1 x 1
1 2 x 2 x3
1 x (1/5) (1/3) (1/5) (1/3) x 1 x
31. lim
lim
x2 2 1x
5
2 1 1/ 2 2 lim x 7 x 3 x x
x 5 x 3 5 x x x
35.
1
x
29.
30. lim
x
1 5 x x2
lim
27.
3
3
8
lim
1 x 12 x 5 2 lim x 7 lim x 7 01 0 x 1 x x 1 x
5
x 2 5 x x3 x 2
1 1x 12 x lim 3 x 8 2 x
1 lim 2 3xx7x x
lim
x2 2 1x
5
3
8
x
1/3
3 x x 4
x
lim
24.
lim
3 x x 4
x
3 2 5 lim 5 x 2 x 5 9 lim 5 x 25 x 49 x , since x n 0, 5x 3 , and the denominator 4. 8
2 lim x 2 x 1 x 8 x 3
34.
x 67 x 3 x
4 x 25 / x
2
lim
x
( x 3)/ x 2
(4 x 25)/ x
2
lim
x
(13/ x ) 4 25/ x
2
(10) 4 0
1 2
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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs 36.
37.
39.
41.
4 3 x 3
lim
lim
6
x
x
x 9
40.
negative positive
42.
positive positive
44. lim
lim x2x8 x 8 4 2 x 7 ( x 7)
4 2/5 x 0 x
4 1/5 2 x 0 ( x )
lim
2
6
3 positive negative
x 3
lim x13
positive positive
lim 2 x3x10 x 5
negative negative
1 2 x 0 x ( x1)
negative positivepositive
lim 21/3 x 0 3 x
(b)
2 lim 1/5 x 0 x
lim
2
x
1 1/3 2 x 0 ( x )
lim
sec x
lim (1 csc )
0
lim (2 cot ) and lim (2 cot ) , so the limit does not exist
0
0
53. (a)
lim 21 x 2 x 4
lim ( x 2)(1 x 2) x 2
(b)
lim 21 x 2 x 4
lim ( x 2)(1 x 2) x 2
(c)
lim 21 x 2 x 4
lim ( x 2)(1 x 2) x 2
(d)
lim 21 x 2 x 4
lim ( x 2)(1 x 2) x 2
54. (a) (b) (c) (d)
55. (a)
lim 2x x 1 x 1 lim 2x x 1 x 1
lim ( x 1)(x x 1) x 1
x 2 x 1 x 1 lim 2x x 1 x 1
lim
lim
x 0
x2 2
lim ( x 1)(x x 1) x 1 lim ( x 1)(x x 1) x 1 lim ( x 1)(x x 1) x 1
1x 0 lim 1x x 0
1 positivepositive
1 positivenegative 1 positivenegative
1 negativenegative
positive positivepositive
positive positivenegative negative positivenegative
negative negativenegative
1 negative
(b)
50.
(0 3) 1 0
lim 25x
48. lim
x 0
1 2/3 x 0 x
lim tan x x
19/ x
47. lim
x
positive negative
lim x 3 2
2 lim 1/5 x 0 x
( x 9)/ x
6
( 4/ x3 3)
38.
x 2
46. (a)
6
lim
positive positive
(43 x3 )/( x3 )
x
lim 21/3 x 0 3 x
52.
x 9 / x
6
lim
lim 31x
45. (a)
51.
6
x 0
43. lim
49.
(4 3 x3 )/ x 6
97
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98
Chapter 2 Limits and Continuity (b) (c) (d)
56. (a) (c) (d)
57. (a) (b) (c)
lim lim lim
x 0
x2 2
3
x 2
x2 2 2
x 2
x 1
22/3 2
1 x
1 x
1 2
2
lim 2xx 14 lim x 1 2 x 1 lim 2 x 4 41
x 1
1 positive
1 21/3 21/3 0 1/3 2
11 23
2
lim 2xx 14 x 2
( x 1)( x 1) 2 x4
positive positive
(b)
2204 0
2 ( x 2)( x1) lim x 33 x 22 lim 2
x 0
lim
x 2
x 2 x
x 0
x 2 3 x 2 x3 2 x 2
positive negative
x ( x 2)
lim
( x 2)( x 1)
x 1 2 x 2 x
lim
x 2 ( x 2)
x 2
negativenegative positivenegative
14 , x 2
2 ( x 2)( x 1) lim x 33 x 22 lim lim x 21 14 , x 2 2
x 2 x 2 x 2
x 3 x 2 3 2 x 2 x 2 x 2
x 3 x 2 3 2 x 0 x 2 x
(e) lim
x 2 x ( x 2) ( x 2)( x 1)
lim lim
lim x 21 14 , x 2
x 2 ( x 2)
x 2
x 2 x
( x 2)( x 1)
x 0
x 2 ( x 2)
x 2 x
negativenegative positivenegative
2 ( x 2)( x 1) ( x 1) 1 1 lim x 3 3 x 2 lim x ( x 2)( x 2) lim x ( x 2) 2(4) 8 x 2 x 4 x x 2 x 2
(b)
2 ( x 2)( x 1) ( x 1) lim x 3 3 x 2 lim x ( x 2)( x 2) lim x ( x 2) x 4 x x 2 x 2 x 2
(c)
2 ( x 2)( x 1) ( x 1) lim x 3 3 x 2 lim x ( x 2)( x 2) lim x ( x 2) x 4 x x 0 x 0 x 0
(d)
x 2 3 x 2 3 x 1 x 4 x
(e)
2
lim 2xx 14 x 2
x 0
(d) lim
58. (a)
1x 0 lim 1x x 0
lim
lim
x 1
( x 2)( x 1) x ( x 2)( x 2)
lim x (xx12)
x 0
and lim x 0
x 1 x ( x 2)
lim
x 1
( x 1) x ( x 2)
negative positivepositive
negative negativepositive
0 (1)(3)
negative negativepositive negative negativepositive
0
so the function has no limit as x 0. 59. (a) 60. (a)
3 lim 2 1/3 t
(b)
1 7 lim 3/5 t
(b)
t 0
t 0
1 lim 2/3 2 2/3 ( x 1) x 1 (c) lim 2/3 2 2/3 ( x 1) x 1 x
61. (a)
62. (a)
x 0
1 lim 1/3 1 4/3 ( x 1) x
x 0
3 lim 2 1/3 t
t 0
1 7 lim 3/5 t
t 0
1 lim 2/3 2 2/3 ( x 1) x 1 (d) lim 2/3 2 2/3 ( x 1) x 1 x
(b)
(b)
x 0
1 lim 1/3 1 4/3 ( x 1) x
x 0
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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs (c)
1 lim 1/3 1 4/3 ( x 1) x
(d)
x 1
1 lim 1/3 1 4/3 ( x 1) x
x 1
63. y
1 x 1
64. y
1 x 1
65. y
1 2 x4
66. y
3 x 3
67. y
x 3 x2
68. y
2x x 1
1 x 1 2
69. domain (, ); y in range and y 4
3x2 , x 2 1
99
2 x21
2 2 0 32x 3 and lim 32x 3 range [4, 7) ; horizontal
x 1
x x 1
asymptote is y 7 .
70. domain (, 1) (1, 1) (1, ); y in range and y lim 22 x x 1 x 1
, and lim 22 x ; x 1 x 1
lim 22 x x 1 x 1
2x ; x 2 1
if x 0, then y 0,
2x 2 x 1 x 1
and lim
lim 22 x x x 1
0,
range (, ); horizontal
asymptote is y 0; vertical asymptotes are x 1, x 1
71. domain (, ); y in range and y x
lim 8e x x 2 e
8 e x 2 e x
x x ; if x ln 8, then y 0, 1 8e x 4, lim 8e x 1, and
2 e
x 2 e
4 range (1, 4); horizontal asymptotes are y 1, y 4
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100
Chapter 2 Limits and Continuity
72. domain (, ); y in range and y x
2x
lim 4ex e2 x x e e
4e x e 2 x e x e2 x
x 2x x 2x , 1 4ex e2 x 4, lim 4ex e2 x 1, and
e e
4 range (1, 4) ; horizontal asymptotes are y 1, y 4
x2 4 , x
73. domain (,0 ) (0, ) ; y in range and y lim
x
x2 4 x
x e e
1; if x 0, then
x2 4 x
1,
lim
x 0
if x 0, then 1
x2 4 x
x2 4 , x
, and lim
x
x2 4 x
lim
x 0
x2 4 x
, and
1
range (, 1) (1, ); horizontal asymptotes are y 1, y 1; vertical asymptote is x 0
74. domain (, 2 ) (2, ); y in range and y x3 x3 8
x3 , lim x3 x 8 x 2 x 3 8 3
x3 x 2 x 8
, lim
3
,
x3 x x 8
lim
1 range (,1) (1, ) ; horizontal asymptote is y 1; vertical asymptote is x 2
75. Here is one possibility.
76. Here is one possibility.
77. Here is one possibility.
78. Here is one possibility.
79. Here is one possibility.
80. Here is one possibility.
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3
1, and
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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs
81. Here is one possibility.
f ( x) x g ( x )
83. Yes. If lim
101
82. Here is one possibility.
f ( x)
2 then the ratio the polynomials’ leading coefficients is 2, so lim g ( x ) 2 as well. x
84. Yes, it can have a horizontal or oblique asymptote. f ( x) x g ( x )
85. At most 1 horizontal asymptote: If lim f ( x) x g ( x )
so lim
86.
x lim
L, then the ratio of the polynomials’ leading coefficients is L,
L as well.
( x 9) ( x 4) x 9 x 4 lim x 9 x 4 x 9 x 4 lim x x 9 4 x x x 9 x 4
lim
x
5 x 9 x 4
5 x
lim
87. lim x 2 25 x 2 1 lim x 2 25 x 2 1 x x 26
lim
x 2 25 x 2 1
x
88.
lim
x
x2
lim
x 1
3 x2
x x2
2
x 25 x 26 x
1 252 1 x
1 x2
2
lim 1 x
( x 2 25) ( x 2 1)
x 2 25 x 2 1
101 0
3x
lim
x 1
3 x2
1
101 0
2 (4 x 2 ) (4 x 2 3 x 2) lim 2 x 4 x 2 3 x 2 lim 2 x 4 x 2 3x 2 2 x 4 x 2 3 x 2 lim 2 x 4 x 3 x 2 x 2 x 4 x 2 3 x 2 x x
3 x 2
lim
x 2 x 4 x 2 3 x 2
lim
x 2
90.
x 2 25 x 2 1
2 ( x 2 3) ( x 2 ) 3 lim x 2 3 x lim x 2 3 x x 2 3 x lim lim x 3 x x x 2 3 x x x x x 2 3 x 3
89.
101 0
1 9x 1 4x
x
3 2x 4 3x 22 x
lim
x
3 x 2 x2
2x x2
4 3x
2
lim
x2
x
3 x 2 x 2x 4 3x 22 x x
3202 34
2 (9 x 2 x ) (9 x 2 ) lim 9 x 2 x 3 x lim 9 x 2 x 3 x 9 x 2 x 3 x lim lim x 9 x x 3 x x 9 x 2 x 3 x x x lim
x
xx
9 x2 x x2 x2
3xx
lim
x
1 9 1x 3
x 9 x 2 x 3 x
313 16
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102
Chapter 2 Limits and Continuity
91. lim x 2 3 x x 2 2 x lim x 2 3x x 2 2 x x x 5x lim lim 2 2 x
x 3 x x 2 x
x2 3 x 5 x 1 3x 1 2x
x 2 x x 2 x lim x 2 x x 2 x x
92. lim
x
2 x 1 1x 1 1x
lim
121 1
x2 3 x x2 2 x
x2 x x2 x x2 x
( x 2 3 x ) ( x 2 2 x )
lim x 2 2 x x 151 52
lim x 2 x x
x 2 3 x x 2 2 x
( x 2 x ) ( x 2 x )
2x
lim
x2 x x2 x
x2 x x2 x
x
93. For any 0, take N 1. Then for all x N we have that f ( x) k k k 0 . 94. For any 0, take N 1. Then for all y N we have that f ( x) k k k 0 . 95. For every real number B 0, we must find a 0 such that for all x, 0 x 0 Now,
1 x2
1 x2
B 0
1 x2
2
B0 x
1 B
x
B so that lim 12 . x0
1 . B
Choose
1 B
B.
, then 0 x x 1
B
x
96. For every real number B 0, we must find a 0 such that for all x, 0 x 0 1 x
1 x2
1 x
B. Now,
B 0 x B1 . Choose B1 . Then 0 x 0 x B1 1 B so that lim 1 . x x0 x
97. For every real number B 0, we must find a 0 such that for all x, 0 x 3 2 ( x 3) 2
2
( x 3) 2 B 0 2 B1 ( x 3)2 ( x 3) 2 2 B 0 so that lim 2 . 2 ( x 3) 2 x 3 ( x 3)
B 0
0 x 3
B2 0
X
3
2 ( x 3) 2
Now,
1 ( x 5)2
x5
B 0 ( x 5)2 1 B
1 ( x 5)2
1 B
x5 1 2 x 5 ( x 5)
B so that lim
1 . Choose B
2, B
1 ( x 5) 2
B.
2 . Choose B
98. For every real number B 0, we must find a 0 such that for all x, 0 x (5)
B. Now, then
1 . Then 0 x (5) B
.
99. (a) We say that f ( x) approaches infinity as x approaches x0 from the left, and write lim f ( x) , x x0
if for every positive number B, there exists a corresponding number 0 such that for all x, x0 x x0 f ( x) B. (b) We say that f ( x) approaches minus infinity as x approaches x0 from the right, and write lim f ( x) , x x0
if for every positive number B (or negative number B) there exists a corresponding number 0 such that for all x, x0 x x0 f ( x ) B. (c) We say that f ( x) approaches minus infinity as x approaches x0 from the left, and write lim f ( x) , if x x0
for every positive number B (or negative number B) there exists a corresponding number 0 such that for all x, x0 x x0 f ( x) B. 100. For B 0,
1 x
B 0 x B1 . Choose B1 . Then 0 x 0 x B1 1x B so that lim x 0
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1 x
.
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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs 101. For B 0, 1x B 0 1x B 0 x 1x B so that lim 1x .
1 B
103
B1 x. Choose B1 . Then x 0 B1 x
x 0
102. For B 0, x 1 2 B x 1 2 B ( x 2) B1 x 2 B1 x 2 B1 . Choose B1 . Then 2 x 2 x 2 0 B1 x 2 0 x 1 2 B 0 so that lim x 1 2 . x 2
B 0 x 2 B1 . Choose B1 . Then 2 x 2 0 x 2 0 x 2 B1 x 1 2 B 0 so that lim x 1 2 .
103. For B 0,
1 x 2
x 2 1, 1 2 1 x
B 1 x 2 B1 (1 x)(1 x) B1 . Now 12x 1 since x 1. Choose 21B . Then 1 x 1 x 1 0 1 x 21B (1 x )(1 x ) B1 12x B1 1 2 B for 0 x 1 and
104. For B 0 and 0 x x near 1
105. y
x2 x 1
107. y
x2 4 x 1
109. y
x 2 1 x
lim 1 2 x 1 1 x
.
106. y
x 2 1 x 1
x 1 x21
x 1 x31
108. y
x 2 1 2 x4
12 x 1 2 x3 4
x 1x
110. y
x3 1 x2
x
x 1 x11
1 x
1 x2
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104
111. y
Chapter 2 Limits and Continuity x
112. y
4 x 2
113. y x 2/3
1 x1/3
1 4 x 2
114. y sin
x 2 1
115. (a) y (see accompanying graph) (b) y (see accompanying graph) (c) cusps at x 1 (see accompanying graph)
116. (a) y 0 and a cusp at x 0 (see the accompanying graph) (b) y 32 (see accompanying graph) (c) a vertical asymptote at x 1 and contains the
point 1,
3 23 4
(see accompanying graph)
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Chapter 2 Practice Exercises
CHAPTER 2
PRACTICE EXERCISES
1. At x 1:
lim f ( x) lim f ( x) 1
x 1
x 1
lim f ( x) 1 f (1) x 1
f is continuous at x 1.
At x 0 :
lim f ( x) lim f ( x) 0
x 0
x 0
lim f ( x) 0. x 0
But f (0) 1 lim f ( x) x 0
f is discontinuous at x 0. If we define f (0) 0, then the discontinuity at x 0 is removable.
At x 1:
lim f ( x) 1 and lim f ( x) 1
x 1
x 1
lim f ( x ) does not exist x 1
f is discontinuous at x 1.
2. At x 1:
lim f ( x) 0 and lim f ( x) 1
x 1
lim f ( x) does not exist
x 1
x 1
f is discontinuous at x 1.
At x 0 :
lim f ( x ) and lim f ( x)
x 0
x 0
lim f ( x) does not exist x 0
f is discontinuous at x 0.
At x 1:
lim f ( x) lim f ( x) 1 lim f ( x) 1.
x 1
x 1
x 1
But f (1) 0 lim f ( x) x 1
f is discontinuous at x 1. If we define f (1) 1, then the discontinuity at x 1 is removable.
3. (a) lim (3 f (t )) 3 lim f (t ) 3( 7) 21 t t0
t t0
2
(b) lim ( f (t )) lim f (t ) (7) 2 49 t t0 t t0 (c) lim ( f (t ) g (t )) lim f (t ) lim g (t ) ( 7)(0) 0 2
t t0
f (t ) t t0 g (t ) 7
(d) lim
t t0 lim f (t )
t t0
lim ( g (t ) 7)
t t0
t t0 lim f (t ) t t0
lim g (t ) lim 7
t t0
t t0
7 07
1
(e) lim cos ( g (t )) cos lim g (t ) cos 0 1 t t0 t t 0
(f)
lim | f (t )| lim f (t ) | 7| 7
t t0
t t0
(g) lim ( f (t ) g (t )) lim f (t ) lim g (t ) 7 0 7 t t0
(h) lim
t t0
1 f (t )
t t0
1
lim f (t )
t t0
t t0
1 7
17
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106
Chapter 2 Limits and Continuity
4. (a) lim g ( x) lim g ( x) 2 x 0
x 0
(b) lim ( g ( x) f ( x)) lim g ( x) lim f ( x) x 0
x 0
x 0
2 12
2 2
(c) lim ( f ( x) g ( x)) lim f ( x) lim g ( x) 12 2 x 0
x 0
x 0
(d) lim f 1( x ) lim 1f ( x ) 11 2 x 0 2 x 0
(e) lim ( x f ( x)) lim x lim f ( x) 0 12 12 x 0 x 0 x 0
(f)
f ( x )cos x x 1
lim
x 0
lim f ( x ) lim cos x
x 0
x 0
lim x lim 1
x0
x 0
12 (1) 1 0 1
2
5. Since lim x 0 we must have that lim (4 g ( x)) 0. Otherwise, if lim (4 g ( x)) is a finite positive number, x 0
x 0
x 0
4 g ( x ) 4 g ( x ) we would have lim x and lim x so the limit could not equal 1 as x 0. Similar x 0 x 0 reasoning holds if lim (4 g ( x)) is a finite negative number. We conclude that lim g ( x) 4. x 0
x 0
6. 2 lim x lim g ( x) lim x lim lim g ( x) 4 lim lim g ( x) 4 lim g ( x) (since lim g ( x) is a x0 x 4 x 0 x 4 x 0 x 0 x 4 x 4 x 0 constant) lim g ( x ) 24 12 . x 0
7. (a) lim f ( x) lim x1/3 c1/3 f (c) for every real number c f is continuous on ( , ). x c
x c
x c
x c
x c
x c
1 c 2/3
x c
x c
1 c1/ 6
(b) lim g ( x) lim x3/4 c3/4 g (c) for every nonnegative real number c g is continuous on [0, ). (c) lim h( x) lim x 2/3 (, ). (d) lim k ( x) lim x 1/6
8. (a)
h(c) for every nonzero real number c h is continuous on (, 0) and k (c ) for every positive real number c k is continuous on (0, )
n 12 , n 12 , where I the set of all integers.
n I
(b)
(n , (n 1) ), where I the set of all integers.
n I
(c) (, ) ( , ) (d) (, 0) (0, ) x2 4 x 4 ( x 2)( x 2) lim x ( x 7)( x 2) lim x (xx27) , x 2; the limit does not exist because 3 2 x 5 x 14 x x 0 x 0 x 0 lim x (xx27) and lim x (xx27) x 0 x 0 x2 4 x 4 ( x 2)( x 2) 0 0 lim 3 2 lim x ( x 7)( x 2) lim x (xx27) , x 2, and lim x (xx27) 2(9) x 5 x 14 x x 2 x2 x 2 x 2
9. (a) lim
(b)
x ( x 1) x2 x lim 3 2 lim 2 x 1 lim 2 1 , x 0 and x 1. 4 3 x 0 x 2 x x x 0 x ( x 2 x 1) x 0 x ( x 1)( x 1) x 0 x ( x 1) 2 Now lim 2 1 and lim 2 1 lim 5 x 4x 3 . x ( x 1) x ( x 1) 2 x x x x 0 x 0 x 0 2 x ( x 1) x x 1 lim 5 4 3 lim 3 2 lim 2 , x 0 and x 1. The limit does not x 1 x 2 x x x 1 x ( x 2 x 1) x 1 x ( x 1) lim 2 1 and lim 2 1 . x 1 x ( x 1) x 1 x ( x 1)
10. (a) lim
(b)
5
exist because
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Chapter 2 Practice Exercises 1 x x 1 1 x
1 x x 1 (1 x )(1 x )
x2 a2 4 4 x a x a
2 2 2 x a ( x a )( x a )
lim
( x h)2 x 2 h x 0
lim
14. lim
1 1 2 x 2
x
x 0
x1/3 1 x 1 x 1
17. lim
lim (2 x h) h
tan 2 x x 0 tan x
19. lim
( x3 6 x 2 12 x 8) 8 x x 0
14
lim ( x 2 6 x 12) 12 x 0
lim
( x 1)( x 1)
2/3 1/3 x 1 ( x 1)( x x 1)
x 1 2/3 1/3 x 1 x x 1
lim
11111
2 3
( x1/3 4)( x1/3 4) ( x1/3 4)( x1/3 4) ( x 2/3 4 x1/3 16)( x 8) lim x 8 x 8 ( x 8)( x 2/3 4 x1/3 16) x 64 x 64 1/3 1/3 ( x 64) ( x 4) ( x 8) ( x 4) ( x 8) (4 4) (88) lim lim 2/3 1/3 161616 83 2/3 1/3 x 64 ( x 64) ( x 4 x 16) x 64 x 4 x 16
lim
sin 2 x cos x x 0 cos 2 x sin x
lim csc x lim
x
1 x 0 4 2 x
lim
lim
x
x
21. lim sin
x 0
( x1/3 1) ( x 2/3 x1/3 1)( x 1) 2/3 1/3 x 1 ( x 1) ( x 1)( x x 1)
x 2/3 16 x 8 x 64
2a
( x 2 2 hx h 2 ) x 2 h x 0
lim
lim
12
h 0
lim
20.
1 2 2 xa x a
lim
lim (2 x h) 2 x
2 (2 x ) x 0 2 x (2 x )
(2 x )3 8 x x 0
12
( x 2 2 hx h 2 ) x 2 h h 0
lim
16. lim
18.
2
( x h)2 x 2 h h 0
13. lim
lim
lim
( x2 a2 )
lim
12. lim
15.
1 x 1 1 x
lim
11. lim
1 sin x
lim
x 0
cos x x 2 x 1 1 1 2 2 sin2 x2 x cos 2 x sin x x
2x sin x sin 2 sin sin 2 1
22. lim cos 2 ( x tan x) cos 2 ( tan ) cos 2 ( ) (1)2 1 x
8x x 0 3sin x x
23. lim
24. lim x 0
8 sin x x 0 3 x 1
lim
cos 2 x 1 sin x
8 3(1) 1
4
sin 2 x 2 x 1 lim cos 2 x 1 lim cossin2xx1 cos cos 2 x 1 x 0 sin x (cos 2 x 1) x 0 sin x (cos 2 x 1) x 0 2
lim
4sin x cos 2 x x 0 cos 2 x 1
lim
4(0)(1) 2 11
2
0
25. Let x = t 3 lim ln(t 3) lim ln x t 3
x 0
26. lim t 2 ln 2 t ln1 0 t 1
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108
Chapter 2 Limits and Continuity
27. 1 cos 1 e1 ecos( / ) e1 e 1 ecos( / ) e lim
0
ecos( / ) 0 by the
Sandwich Theorem 28.
lim
2e1/ z
z 0 e1/ z
1
2
lim
120 2
1/ z
z 1 e
1/3
29.
lim [4 g ( x)]1/3 2 lim 4 g ( x) x 0 x 0
30.
1 x g ( x) x 5 3 x 2 1 x 1 g ( x )
32.
x 0
x 0
2 lim ( x g ( x)) 12 5 lim g ( x)
lim
31. lim
2 lim 4 g ( x) 8, since 23 8. Then lim g ( x ) 2.
x 5
x 5
1 2
lim g ( x) 12 5 x 5
lim g ( x) 0 since lim (3 x 2 1) 4 x 1
5 x 2 x 2 g ( x )
x 1
0 lim g ( x) since lim (5 x 2 ) 1
lim
x 2
x 2
33. (a) f (1) 1 and f (2) 5 f has a root between 1 and 2 by the Intermediate Value Theorem. (b), (c) root is 1.32471795724 34. (a) f (2) 2 and f (0) 2 f has a root between 2 and 0 by the Intermediate Value Theorem. (b), (c) root is 1.76929235424 35. At x 1:
lim
lim
x 1 x ( x 2 1) x 2 1
x 1
lim
x 1
x ( x 2 1)
f ( x) lim
2 x 1 | x 1|
lim x 1, and x 1
f ( x) lim
x ( x 2 1)
x 1
| x 2 1|
lim
lim ( x) ( 1) 1. Since lim x 1
lim
x 1
x ( x 2 1)
2 x 1 ( x 1)
x 1
f ( x)
f ( x) lim f ( x) does not exist, the x1
function f cannot be extended to a continuous function at x 1. At x 1:
lim f ( x) lim
x 1
x ( x 2 1) 2
x 1 | x 1| x ( x 2 1)
lim
x 1
x 2 1
lim
x ( x 2 1) 2
x 1 ( x 1)
lim ( x) 1, and lim f ( x) lim x 1
x 1
x ( x 2 1)
2 x 1 | x 1|
lim x 1. x 1
Again lim f ( x) does not exist so f cannot be extended to a continuous function at x 1 either. x 1
36. The discontinuity at x 0 of f ( x) sin
lim sin 1x does not exist. 1x is nonremovable because x 0
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Chapter 2 Practice Exercises 37. Yes, f does have a continuous extension at a 1: define f (1) lim x 41 43 . x 1 x x
38. Yes, g does have a continuous extension at a 2 :
5 cos g 2 lim 4 2 54 . 2
39. From the graph we see that lim h(t ) lim h(t ) t 0
t 0
so h cannot be extended to a continuous function at a 0.
40. From the graph we see that lim k ( x) lim k ( x) x 0
x 0
so k cannot be extended to a continuous function at a 0.
41.
43.
3
2 lim 52xx73 lim 7x 5200 52 x x 5 x 2 lim x 43x 8 lim
x
3x
1 x 3 x
42.
2 lim 2 x2 3 lim
x 5 x 7
x
2
3
20
x2 5 72 x
5 0 52
42 8 3 0 0 0 0 3x
3x
1
44.
45.
lim 2 1 x x 7 x 1
x2 7 1 1 x x 2 x
lim
100 0 0
2 lim x x71 x lim x 17 x x 1 x
46.
x 4 x3 3 x 12 x 128
lim
x 1 128 x 12 3
lim
x
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109
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110 47. 48.
Chapter 2 Limits and Continuity lim sin x x x
lim 1 0 since x as x lim sin x 0. x x x x
cos 1
lim
lim 2 0 lim cos 1 0. sin x
2 x
49.
1 x x2 x lim x sin lim sin x x x sin x x 1 x
50.
x 2/3 x 1 lim 1 x 5/3 1 0 1 lim 2/3 2 2 x x cos x x 1 cos x 1 0 x2/3
51. 52.
53.
54.
1100 0 1
lim e1/ x cos 1x e0 cos(0) 1 1 1
x
lim ln 1 1t ln1 0
x
lim tan 1 x 2
x
lim e3t sin 1 1t 0 sin 1 (0) 0 0 0
t
2 2 is undefined at x 3 : lim xx 34 and lim xx 34 , thus x 3 is a vertical asymptote. x 3 x 3 2 2 2 (b) y x2 x 2 is undefined at x 1: lim x2 x 2 and lim x2 x 2 , thus x 1 is a vertical
55. (a) y
x2 4 x 3
x 2 x 1
x 1 x 2 x 1
x 1 x 2 x 1
asymptote. 2 (c) y x2 x 6 is undefined at x 2 and 4: lim
x 2 x 6 2 x 2 x 2 x 8
2 lim xx34 56 ; lim x2 x 6 lim xx34 x 2 x 8 x2 x 4 x 2 x 8 x 4 2 x x 6 x 3 lim 2 lim x 4 . Thus x 4 is a vertical asymptote.
x 4 x 2 x 8
56. (a)
x 4
2
1 x 2 x 1 x x 2 1
y 12 x : lim
1
x2
lim
1
1 x 1 2 x
1 x 2 2 x x 1
1 1
1 and lim
1
x2
lim
x 1
1 1 x2
1 1
1, thus y 1 is a
horizontal asymptote. (b)
y
(c) y
x 4 : x4 x2 4 x
lim
x
:
lim
x
x 4 x4
lim
x2 4 x
1
4 x
lim
x
lim
x
1 0 1 0
x 1 4x
4
1
x2
1
1
1, thus y 1 is a horizontal asymptote.
1 0 1
1 and lim
x
4
x2
x x
lim
x
1
4
x2
1
1 0 1
x2 4 x
1 1
lim
1
x
4
x2
x x2
1,
thus y 1 and y 1 are horizontal asymptotes.
(d) y
x 2 9 : 9 x 2 1
lim
x
x 2 9 9 x 2 1
lim
x
1 9
thus y 13 is a horizontal asymptote.
9
x2 1 x2
1 0 9 0
1 3
and lim
x
x 2 9 9 x 2 1
lim
x
1
9
x2 9 12 x
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1 0 9 0
13 ,
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Chapter 2 Additional and Advanced Exercises 57. domain [4, 2) (2, 4]; y in range and y 16 x 2 x2
lim
x 2
ax 2 4 x b
x b
x x x b
a
x x x b
lim
CHAPTER 2
1. (a) x x
lim
x2
16 x 2 x 2
, and
4 x2
a
ax 2 4 x x b
vertical asymptote is x b; lim a horizontal asymptote is y a ,
4 x2
x x x b
lim
a
4 x2
lim
x
x x x b
lim
a
4 x2
ax 2 4 x b
a horizontal asymptote is y a
ADDITIONAL AND ADVANCED EXERCISES 0.1
x
, if x 4, then y 0,
, range (, )
58. Since lim lim
16 x 2 x 2
0.01
0.001
0.0001 0.00001
0.7943 0.9550 0.9931 0.9991 0.9999
Apparently, lim x x 1 x 0
(b)
2. (a) x
10
1x
1/(ln x )
1000
0.3679 0.3679 0.3679
1 x x
Apparently, lim (b)
100
1/(ln x )
0.3678 1e
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111
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112
Chapter 2 Limits and Continuity
3.
v2 c2
lim L lim L0 1
v c
v c
L0 1
lim v 2
v c 2
c
2
L0 1 c 2 0 c
The left-hand limit was needed because the function L is undefined if v c (the rocket cannot move faster than the speed of light). 4. (a)
x 2
1 0.2 0.2 2x 1 0.2 0.8 2x 1.2 1.6 x 2.4 2.56 x 5.76.
(b)
x 2
1 0.1 0.1 2x 1 0.1 0.9 2x 1.1 1.8 x 2.2 3.24 x 4.84.
5. |10 (t 70) 104 10| 0.0005 |(t 70) 104 | 0.0005 0.0005 (t 70) 104 0.0005 5 t 70 5 65 t 75 Within 5 F. 6. We want to know in what interval to hold values of h to make V satisfy the inequality |V 1000| |36 h 1000| 10. To find out, we solve the inequality: 990 h 1010 8.8 h 8.9 |36 h 1000| 10 10 36 h 1000 10 990 36 h 1010 36 36 where 8.8 was rounded up, to be safe, and 8.9 was rounded down, to be safe. The interval in which we should hold h is about 8.9 8.8 0.1 cm wide (1 mm). With stripes 1 mm wide, we can expect to measure a liter of water with an accuracy of 1%, which is more than enough accuracy for cooking. 7. Show lim f ( x) lim ( x 2 7) 6 f (1). x 1
x 1
Step 1: |( x 2 7) 6| x 2 1 1 x 2 1 1 x 1 . Step 2: | x 1| x 1 1 x 1. Then 1 1 or 1 1 . Choose min 1 1 , 1 1 , then 0 | x 1|
|( x 2 7) 6| and lim f ( x) 6. By the continuity text, f ( x ) is continuous at x 1. x 1
1 x 14 2 x
8. Show lim g ( x) lim Step
x 14 1: 21x
2 g 14 .
2 21x 2 2 21x 2 412 x 412 .
14 x 14 14 x 14 . Then 14 412 14 412 4(2 ) , or 14 412
Step 2:
X
1 4 2
14 4(2 ) . Choose 4(2 ) , the smaller of the two values. Then 0 x 14 21x 2 and lim 21x 2. x 1 4
By the continuity test, g ( x) is continuous at x 14 . 9. Show lim h( x) lim 2 x 3 1 h(2). x 2
Step 1:
x2
2 x 3 1 2 x 3 1 1 2 x 3 1
Step 2: | x 2| x 2 or 2 x 2.
(1 )2 3 2
x
(1 )2 3 . 2
2 2 2 (1 ) 2 3 (1 ) 2 3 1(1 )2 2 , or 2 (1 ) 3 (1 ) 3 2 (1 ) 1 2 2 2 2 2 2 2 2 2 2 2 . Choose 2 , the smaller of the two values. Then, 0 | x 2| 2 x 3 1 ,
Then 2
so lim 2 x 3 1. By the continuity test, h( x) is continuous at x 2. x2
10. Show lim F ( x) lim 9 x 2 F (5). x 5
Step 1:
x 5
9 x 2 9 x 2 9 (2 )2 x 9 (2 ) 2 .
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Chapter 2 Additional and Advanced Exercises
113
Step 2: 0 | x 5| x 5 5 x 5. Then 5 9 (2 )2 (2 )2 4 2 2, or 5 9 (2 ) 2 4 (2 ) 2 2 2. Choose 2 2, the smaller of the two values. Then, 0 | x 5| 9 x 2 , so lim 9 x 2. x 5
By the continuity test, F ( x) is continuous at x 5.
11. Suppose L1 and L2 are two different limits. Without loss of generality assume L2 L1. Let 13 ( L2 L1 ). Since lim f ( x) L1 there is a 1 0 such that 0 | x x0 | 1 | f ( x ) L1 | f ( x ) L1 x x0
13 ( L2 L1 ) L1 f ( x) 13 ( L2 L1 ) L1 4 L1 L2 3 f ( x) 2 L1 L2 . Likewise, lim f ( x) L2 so x x0
there is a 2 such that 0 | x x0 | 2 | f ( x ) L2 | f ( x) L2 13 ( L2 L1 ) L2 f ( x) 13 ( L2 L1 ) L2 2 L2 L1 3 f ( x) 4 L2 L1 L1 4 L2 3 f ( x) 2 L2 L1. If min{1 , 2 } both inequalities must 4 L L 3 f ( x) 2 L1 L2 hold for 0 | x x0 | : 1 2 5( L1 L2 ) 0 L1 L2 . That is, L1 L2 0 and L1 4 L2 3 f ( x) 2 L2 L1 L1 L2 0, a contradiction. 12. Suppose lim f ( x) L. If k 0, then lim k f ( x) lim 0 0 0 lim f ( x) and we are done. If k 0, then given x c
x c
x c
x c
any 0, there is a 0 so that 0 | x c | | f ( x) L | | k | | k || f ( x ) L | | k ( f ( x) L)|
|(kf ( x)) (kL)| . Thus lim k f ( x) kL k lim f ( x) . x c x c
13. (a) Since x 0 , 0 x3 x 1 ( x3 x) 0 lim f ( x3 x) lim f ( y ) B where y x3 x.
x 0
y 0
(b) Since x 0 , 1 x x 0 ( x x) 0 lim f ( x x) lim f ( y ) A where y x3 x. 3
3
x 0
3
y 0
(c) Since x 0 , 0 x x 1 ( x x ) 0 lim f ( x x ) lim f ( y ) A where y x 2 x 4 . 4
2
2
4
x 0 4
2
4
y 0 2
(d) Since x 0 , 1 x 0 0 x x 1 ( x x ) 0 lim f ( x x 4 ) A as in part (c). 4
2
2
x 0
14. (a) True, because if lim ( f ( x) g ( x)) exists then lim ( f ( x) g ( x)) lim f ( x) lim [( f ( x) g ( x)) f ( x)] x a
lim g ( x) exists, contrary to assumption.
x a
(b) False; for example take f ( x)
1 x
x a
x a
x a
and g ( x) 1x . Then neither lim f ( x) nor lim g ( x) exists, but x 0
x 0
lim ( f ( x) g ( x)) lim 1x 1x lim 0 0 exists. x 0 x 0 x 0 (c) True, because g ( x) | x | is continuous g ( f ( x)) | f ( x)| is continuous (it is the composite of continuous functions). 1, x 0 (d) False; for example let f ( x) f ( x) is discontinuous at x 0. However | f ( x)| 1 is 1, x 0 continuous at x 0. x 2 1 x 1 x 1
15. Show lim f ( x ) lim x 1
lim
x 1
( x 1)( x 1) ( x 1)
2, x 1.
x 2 1 , x 1 . We now prove the limit of f ( x) as x 1 Define the continuous extension of f ( x) as F ( x ) x 1 2 , x 1
exists and has the correct value. Step 1:
x 2 1 ( 2) x 1
( x 1)( x 1) ( x 1)
2 ( x 1) 2 , x 1 1 x 1.
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114
Chapter 2 Limits and Continuity Then 1 1 , or 1 1 . Choose . Then 0 | x (1)| 2 xx 11 (2) lim F ( x) 2. Since the conditions of the continuity test are met by F ( x), then f ( x) has x 1 a continuous extension to F ( x) at x 1.
x 2 2 x 3 x 3 2 x 6
16. Show lim g ( x) lim x 3
( x 3)( x 1) x 3 2( x 3)
lim
2, x 3.
x 2 2 x 3 , x 3 . We now prove the limit of g ( x) as x 3 Define the continuous extension of g ( x) as G ( x) 2 x 6 , x3 2 exists and has the correct value.
Step 1:
x 2 2 x 3 2 2 x 6
( x 3)( x 1) 2( x 3)
2 x21 2 , x 3 3 2 x 3 2.
Step 2: | x 3| x 3 3 x 3. Then, 3 3 2 2, or 3 3 2 2. Choose 2. Then 0 | x 3|
2 ( x 3)( x 1) x 2x2x63 2 lim 2( x 3) 2. Since the conditions of the continuity test hold for G ( x), g ( x) can be x 3 continuously extended to G ( x ) at x 3.
17. (a) Let 0 be given. If x is rational, then f ( x) x | f ( x) 0| | x 0| | x 0| ; i.e., choose . Then | x 0| | f ( x) 0| for x rational. If x is irrational, then f ( x) 0 | f ( x) 0| 0 which is true no matter how close irrational x is to 0, so again we can choose . In either case, given 0 there is a 0 such that 0 | x 0| | f ( x) 0| . Therefore, f is continuous at x 0. (b) Choose x c 0. Then within any interval (c , c ) there are both rational and irrational numbers. If c is rational, pick 2c . No matter how small we choose 0 there is an irrational number x in (c , c ) | f ( x) f (c)| |0 c | c 2c . That is, f is not continuous at any rational c 0. On the other hand, suppose c is irrational f (c) 0. Again pick 2c . No matter how small we choose 0 there is a rational number x in (c , c ) with | x c | 2c 2c x 32c . Then | f ( x) f (c)| | x 0| | x|
c 2
f is not continuous at any irrational c 0.
If x c 0, repeat the argument picking value x c.
|c| 2
2c . Therefore f fails to be continuous at any nonzero
18. (a) Let c mn be a rational number in [0, 1] reduced to lowest terms f (c) 1n . Pick 21n . No matter how small 0 is taken, there is an irrational number x in the interval (c , c ) | f ( x) f (c)| 0 1n 1n 21n . Therefore f is discontinuous at x c, a rational number.
(b) Now suppose c is an irrational number f (c) 0. Let 0 be given. Notice that 12 is the only rational number reduced to lowest terms with denominator 2 and belonging to [0, 1]; 13 and 23 the only rationals with denominator 3 belonging to [0, 1]; 14 and 34 with denominator 4 in [0, 1]; 15 , 52 , 53 and 54 with denominator 5 in [0, 1]; etc. In general, choose N so that
1 N
there exist only finitely many rationals in [0, 1] having
denominator N , say r1 , r2 , , rp . Let min {| c ri |: i 1, , p}. Then the interval (c , c ) contains no rational numbers with denominator N . Thus, 0 | x c | | f ( x) f (c)| | f ( x ) 0| | f ( x)| N1 f is continuous at x c irrational.
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Chapter 2 Additional and Advanced Exercises
115
(c) The graph looks like the markings on a typical ruler when the points ( x, f ( x)) on the graph of f ( x ) are connected to the x-axis with vertical lines.
19. Yes. Let R be the radius of the equator (earth) and suppose at a fixed instant of time we label noon as the zero point, 0, on the equator 0 R represents the midnight point (at the same exact time). Suppose x1 is a point on the equator “just after” noon x1 R is simultaneously “just after” midnight. It seems reasonable that the temperature T at a point just after noon is hotter than it would be at the diametrically opposite point just after midnight: That is, T ( x1 ) T ( x1 R ) 0. At exactly the same moment in time pick x2 to be a point just before midnight x2 R is just before noon. Then T ( x2 ) T ( x2 R ) 0. Assuming the temperature function T is continuous along the equator (which is reasonable), the Intermediate Value Theorem says there is a point c between 0 (noon) and R (simultaneously midnight) such that T (c) T (c R ) 0; i.e., there is always a pair of antipodal points on the earth’s equator where the temperatures are the same. 1 ( f ( x ) g ( x )) 2 x c 4 1 (32 ( 1) 2 ) 2. 4
20. lim f ( x ) g ( x) lim x c
( f ( x) g ( x))2
(b) At x
2
( x ) g ( x)) lim ( f ( x) g ( x)) x c
2
1(1 a ) 1 1 a lim 1 a1 a 1 1 a lim 1 12 a 1 1 a a 1 1 0 ( 1 1 a ) a 0 a 0 a 0 a 0 1(1 a ) a 1 1: lim r (a ) lim lim 1 1 0 a 1 a 1 a ( 1 1 a ) a 1 a ( 1 1 a ) 1 (1 a ) a 0: lim r (a ) lim 1 a1 a lim 1 a1 a 1 1 a lim lim a ( 1 1 a ) a 1 1 a 0 a 0 a 0 a 0 a 0 a ( 1 1 a ) 1 lim (because the denominator is always negative); lim r (a) a 0 1 1 a a 0 1 lim (because the denominator is always positive). a 0 1 1 a
21. (a) At x 0: lim r (a ) lim At x
1 lim ( f 4 x c
Therefore, lim r (a ) does not exist. a 0
At x 1: lim r (a ) lim a 1
a 1
1 1 a a
1 a 1 1 1 a
lim
1
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116
Chapter 2 Limits and Continuity (c)
(d)
22. f ( x) x 2 cos x f (0) 0 2 cos 0 2 0 and f ( ) 2 cos( ) 2 0. Since f ( x ) is continuous on [ , 0], by the Intermediate Value Theorem, f ( x ) must take on every value between [ 2, 2]. Thus there is some number c in [ , 0] such that f (c) 0; i.e., c is a solution to x 2 cos x 0. 23. (a) The function f is bounded on D if f ( x) M and f ( x) N for all x in D. This means M f ( x) N for all x in D. Choose B to be max {| M |, | N |}. Then | f ( x)| B. On the other hand, if | f ( x)| B, then B f ( x) B f ( x) B and f ( x) B f ( x) is bounded on D with N B an upper bound and M B a lower bound. (b) Assume f ( x) N for all x and that L N . Let L 2N . Since lim f ( x) L there is a 0 such that x x0
0 | x x0 | | f ( x) L | L f ( x ) L L L 2N f ( x) L L 2N L 2N f ( x) 3 L2 N . But L N L 2N N N f ( x) contrary to the boundedness assumption f ( x) N . This contradiction proves L N . (c) Assume M f ( x) for all x and that L M . Let M2 L . As in part (b), 0 | x x0 | L M2 L f ( x) L M2 L 3L 2 M f ( x)
M L 2
M , a contradiction.
25.
lim
x 0
sin(1cos x ) x
sin(1cos x ) 1cos x 1 cos x x 1 cos x x 0 1cos x
lim
a b 2
| a b |
2 a 2 b a 2 b 22a a. |a b| If a b, then a b 0 | a b | (a b) b a max {a, b} a 2 b 2 a 2 b b 2 a 22b b. | a b | (b) Let min {a, b} a 2 b 2 .
24. (a) If a b, then a b 0 | a b | a b max {a, b}
2 sin(1cos x ) x lim x1(1cos x x) 1 cos cos x 0 x 0
lim
2 x lim sin x sin x 1 0 0. 1 lim x (1sin x ) x 0 x cos 1 cos x 2 x 0
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Chapter 2 Additional and Advanced Exercises 26.
lim sin x x 0 sin x
lim
x 0
sin(sin x ) x
lim
sin( x 2 x ) x x 0
lim
sin( x 2 4) x2 x 2
lim
sin( x 3) x 9 x 9
lim
27. lim
x 0
x sin x x sin x
x 0
28. lim
29. lim
x x
sin(sin x ) sin x x sin x sin( x 2 x ) 2
x x
x 0
30. lim
sin( x 2 4) 2
x 4
x 2
1 lim
x 0
lim
x 0
lim
x 1 1 0 0.
x 0
sin(sin x ) lim sinx x sin x x 0
( x 1) lim
sin( x 2 x )
x2 x
x 0
sin( x 2 4)
( x 2) lim
x2 4
x2
sin( x 3) 1 x 3 x 3 x 9
1 sin x x
1 1 1.
lim ( x 1) 1 1 1. x 0
lim ( x 2) 1 4 4. x2
sin( x 3) lim x 3 x 9 x 9
lim
117
1 x 3
1 16 16 .
31. Since the highest power of x in the numerator is 1 more than the highest power of x in the denominator, there is 3/ 2 an oblique asymptote. y 2 x 2 x 3 2 x 3 , thus the oblique asymptote is y 2 x. x 1
32. As x ,
1 x
x 1
x; thus
0 sin 1x 0 1 sin 1x 1, thus as x , y x x sin 1x x 1 sin 1x
the oblique asymptote is y x.
33. As x , x 2 1 x 2 x 2 1 x 2 ; as x , x 2 x, and as x , x 2 x; thus the oblique asymptotes are y x and y x. 34. As x , x 2 x x 2 2 x x( x 2) x 2 ; as x , x 2 x, and as x , x 2 x; asymptotes are y x and y x. 35. Assume 1 a b and
a x
x
1 x b
a ( x b) x 2 ( x b) x f ( x) a( x b) x 2 ( x b) x 0; f is
continuous for all x-values and f (0) ab 0, f (a b) a 2 (a b)2 a (a b) a2 a (a b) 2 a b 0. ()
()
Thus, by the Intermediate Value Theorem there is at least one number c, 0 c a b, so that f (c) 0 a(c b) c 2 (c b) c 0 ac c c 1b .
36. (a)
lim
x 0
a bx 1 x
" a01" , so
1bx 1 1bx 1 x 1bx 1 x 0
a 1 0 a 1, then lim
b b 2b4 x 0 1 bx 1 2 tan( ax a ) b 2 "tan 0 b 2" "b 2" lim 0 , x 1 0 x 1
(1bx ) 1 ( 1bx 1) x x 0
lim
lim
(b)
tan( ax a ) x 1 x 1
so b 2 0 b 2, then lim
sin a ( x 1)
lim cos aa( x 1) a ( x 1) cosa 0 1 a 3 x 1
37.
tan a ( x 1)
lim a a ( x 1) x 1
1/6 4 4 1/6 1/6 1/6 2 1/6 1/3 2/3 1 lim ( x ) 1 lim ( x 1)( x 1)(( x ) 1) lim ( x 1)( x 1) (2)(2) 4 lim x 1/2 1/6 1/6 1/6 2 1/6 1/3 3 1/6 3 3 3 x 1 1 x x 1 1 ( x ) x 1 (1 x )(1 ( x ) ( x ) ) x 1 1 x x
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118 38.
Chapter 2 Limits and Continuity lim
3x 4 x 4 x
lim
4x x
x 0
x 0
lim x 0
(3 x 4) x 4 x
3x 4 x 4 x x 0
4 lim
3x 4 x 4 (3 x 4) ( x ) 4 lim 22x 2; assume x 43 lim lim x x x 0 x 0 x 0
does not exist.
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