2 2路1
Special limits 1. lim x l3
1 x 3 x 3 1 lim lim x l 3 x l 3 ( x 3)( x 4 ) x 4 7 x x 12 2
( x h)2 x 2 x 2 2 xh h 2 x 2 2 xh h 2 lim(2 x h) 2 x lim lim hl 0 hl 0 hl 0 hl 0 h h h
2. lim
( x 4 )( x 2 4 x 16) x 3 64 x 2 4 x 16 16 16 16 lim 6 lim 2 x l 4 x 16 x l4 x l4 ( x 4 )( x 4)) x 4 8 f ( x h) f ( x ) (5( x h) 8) (5 x 8) 5h lim 4. If f ( x ) 5 x 8, lim lim lim 5 5 hl 0 h l 0 h l 0 h h h hl 0 5 x 7 15 7 8 4 5. lim 2 x l 3 x 3 9 3 6 3 3. lim
1 x 5 x 25 1 lim lim 6. xlim l 25 x 25 x l 25 x l 25 ( x 25)( x 5) x 5 10 g ( x ) g ( 2) x2 4 ( x 2)( x 2) lim lim lim( x 2) 4 x l2 x l 2 x l 2 x l2 x 2 x 2 x 2 2 2x 4 x x (2 x 4 ) 8. lim lim lim(2 x 4 ) 4 x l0 x l 0 x l0 x x 7. If g ( x ) x 2 , lim
( x r x )( x r x ) x r x (x r ) x 1 1 lim lim lim r l0 r l0 r l0 r r( x r x ) r( x r x ) ( x r x) 2 x
9. lim r l0
x3 6 This yields 0 in the denominator and 70 in the numerator, so the limit does not exist. x l4 x 4
10. lim
2路2
1. lim 5 x 7 c x lc
2. lim
x lc
7 0 x3
3. lim 3x 95 c x l c
x 3 x 2 47 x 9 4. lim lim x lc 18 x 3 76 x 11 x lc 5. lim
x lc
1 47 9 x x2 x3 1 0 0 0 1 76 11 18 0 0 18 18 2 3 x x
1
8 0 4 x
1 2 2 0 0 x 2 0 6. lim 2 lim x x x lc x 5 x 6 x lc 5 6 1 0 0 1 2 x x 1 6 7 x 5 6x 3 7 x x3 x6 0 0 0 0 7. lim 6 lim x lc 5 x 6 x 2 11 x lc 6 11 5 0 0 5 4 6 x x 7 x 4 6 x 2 3x lim 8. lim x l c 3x 3 7 x 5 x l c
118
Worked solutions
6 3 x x2 c 7 5 3 2 3 x x 7x