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First Edition

The Sky is the Limit

Maggie Reisdorf


The Sky is the Limit

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Maggie Reisdorf W200 - Spring 2014

MA.C.1.1 2000 Understand the concept of limit and estimate limits from graphs and tables of values.


Directions

How To Use This Book For Starters ! This is an interactive e-book! The purpose of this is for you to learn the introduction to limits in calculus in a more hands-on way. There will be short questions to track your progress and help you stay engaged. There are widgets from iBook Author and Bookry. Click on any of the icons to interact or simply answer a question right on the page! ! Here we will do a practice problem that should be simple just to get the hang of the system!

Practice 1: Geometry Which of these shapes has 4 sides and a name that starts with an R?

A. Circle B. Triangle C. Rectangle D. Square

Check Answer

! Hopefully you got the right answer, but don’t worry if you didn’t! This book will focus more on the basic topic of limits in calculus.

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Limits

What is a limit?

! For example, in the graph below, as you follow the line and the yellow arrow from the left, you get closer to x=1 and you come close to hitting y=2.

What is a limit? ! Many times in calculus, there are some concepts or items that cannot be attacked, or solved, head-on. As you work your way through calculus, studying limits, derivatives, integrals, and more, you will find this to be very true. You are familiar with holes in graphs. In some situations, you may want to evaluate what the value would have been if there wasn’t a hole. To do this, we will use limits.

Click on the icon to view how limits interact with a specific x-value.

! ! From the right side, as you follow the line and the pink arrow, the x-values are decreasing, coming closer to x=1 and still, Â you draw nearer to y=2. Based on this information, we can say that the limit of this function as it approaches 1 is 2. ! 3


! Now let’s look at the notation for limits! The problem we are looking at is: What is the limit of f(x)=(x^2-1)/(x-1) as x approaches 1? Now, this is our notation!

Components of the limit notation

! First we explained the limit as drawing nearer to a spot. Expand your understanding of a limit to another synonymous word approaching. When a limit is approaching a value, we can then determine a numerical limit! ! Graphs aren’t always easily accessible and often times in math you simply have an equation with which you can work. Take the same equation from the first page! y=(x^2-1)/(x-1) Here we will evaluate the limit from both the right and left sides. Starting from the left side, you will create a table of values.

Finding limits by using tables of values

! When you create the xvalues on your own, you will first pick an x-value that is close to, but less than, 1 - such as .5. Pick another few x-values that are progressively closer to 1. Then, evaluate their corresponding y-values. Let’s take a look at an example table (on the left) to Click on the icon to solve for yevaluate the limit. The x-values values of our first equation! are provided for you. Just calculate the y-values!

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! From that table, we can see that although the right side of the table or the y-values will never be exactly 2, that is what it is getting closer to. With this information, we can say that the limit as x is approaching 1 from the left is 2.

! Now we will do the same thing from the right hand side. This shows the same procedure on deciding x-values. And once again, these values are placed in a table for you to evaluate the corresponding y-values. Now, fill in the y-values!

!

Finding limits using tables of values

! To indicate that this limit is being evaluated from the left, we will use this notation. The negative sign shows that it is approaching from lesser values than the x-value we are approaching. Click on the icon to calculate a new set of y-values!

! Again you can see that these y-values get closer and closer to 2, but never actually hit the number. From here we can say that the limit as x is approaching 1 from the right is 2. !

!

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! The notation for “the limit as x is approaching 1 from the right is 2� is shown here.

! But what happens when the limits from the left and the right are different? Take a look at the graph of f(x) below. For this graphical analysis, it is not necessary to know the equation, so we will look at this more as an image.

! The plus sign indicates that the limit is being evaluated from values greater than our specific x-value. ! ! When we see that the limits from the left and the right are the same, we can say generally that the limit as x approaches 1 is 2.

! This is where we bring back to the original notation, without a plus or minus sign next to the x-value.

! From the left side of the graph, as x approaches 1, the yvalue is getting closer to and nearly reaching 3. Looking from the right, the x-value approaches 1 and the y-value approaches 1. So here the limit from the left is 3, but from the right is 1. What is our result?

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! Since the two are unequivocal, the limit does not exist. Cady Herron faced a similar issue in a state math competition in the movie Mean Girls. Check it out!

Mean Girls (2004) The limit does not exist!

Watch Cady Herron from Mean Girls solve a limit problem at the state math finals. It can be uncomfortable to say that the limit DNE, but sometimes that is the case!

!

! Limits do not always have to be evaluated with equations as complex as seen in the Mean Girls clip. Simple functions have limits that can be evaluated as well! Many times, limits can be found by plugging in the original x-value! For instance, look at the graph of f(x)=sinx.

! To evaluate the limit of f(x) as x approaches π, you do not need to make tables! Just plug in π for x into the equation! Sinπ=0. Or, you can find the y-value from the graph. Either way, the limit as x approaches π is 0! Use the graph or calculate for confirmation.

! Be careful! The limit from each side does exist, but the general limit as x approaches 1 does not exist. This can be abbreviated as DNE. 7


! Now we will take a look at functions that continue on forever! These are equations where we evaluate the limit as x approaches ∞ (infinity). What does this really mean? In these cases, we look at the long term, as far out as possible to see what the equation could possibly be doing. In application problems, we often see time as the x-axis. Evaluating the limit as x approaches infinity is similar to saying what will be happening at the very end of time.

! Let’s look at the equation f(x)=1/x. We know that this graph has a vertical asymptote at x=0 and a horizontal asymptote at y=0. Fill out the table of values for this equation and try to understand the pattern of y-values.

Finding limits using tables of values

Application Honeybees

Click on the icon to calculate the y-values for f(x)=1/x.

Click through the images to understand how limits apply to honeybees.

! We see that as our x-values get much larger, the y-values get much closer to 0. Because you can never have a 0 in the denominator, we will never reach a concrete point and stop there. In this equation, the numbers just keep inching their way closer to 0. 8


Examples Khan Academy

! There are tons of resources online for free! Make sure you are choosing reliable, accurate resources. Sal Khan provides incredible videos such as the one you have already seen. Videos from Khan Academy are great ways to refresh your memory from class lessons, clear up questions, or completely learn a new topic! Khan has lots to say about limits. Navigate this page to check out the options!

Khan Academy Watch three great examples from Sal Khan about evaluating limits. He shows a few good tricks in this video. This will help you solve future limit problems!

! Once you watch the video, think about the “tricks” he uses. Don’t be afraid to use your algebra skills. Canceling an element can make evaluating a limit much easier! Also, it is completely acceptable on homework or an exam to write your explanations out in full sentences. You don’t always have to have an equation to prove the limit!

Click on the icon to navigate to the Khan Academy page for limits in calculus. Check it out or view some new examples.

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Review/Application Questions

1. Calculate the limit

2. Calculate the limit

Using the equation f(x)=3x/4, find the limit as x approaches 2. Simplify your answer.

x lim 2 x→ 3 + x − 2 x − 3

A. 5/4

A. ∞

B. 3

B. -∞

C. 3/2

C. 0

D. DNE

D. DNE

Check Answer

Check Answer

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3. Calculate the limit

4. Select the answer that best fits. This statement is...

A. 1/2 B. 0 C. ∞ D. DNE

Check Answer

A. Always true B. Sometimes true C. Never true (False)

Check Answer

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About the Author Maggie Reisdorf

! My name is Maggie Reisdorf and I am a student at Indiana University Bloomington. I am an aspiring calculus teacher and my favorite topic is the introduction to derivatives. The topic of limits is a great place to start for this because derivatives are defined by limits! Â

! I am originally from Mishawaka, IN where I graduated from Mishawaka High School in 2013. I found my love of math at an early age. As a baby, my dad counted aloud to get me to calm down rather than shushing me. I have a great passion for students and children. I am thrilled to apply this passion in the future!

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Resources 1. Graph with hole at x=1: http://www.math-prof.com/Alg2/Alg2_Ch_23.asp 2. Man looking in hole: http://tinyurl.com/njg9yn7 3. Limit Notation at 1: http://www.wolframalpha.com/input/?i=the+limit+of+f%28x%29%3D%28x%5E2-1%29%2F%28x-1%29+as+x+approaches+1+ 4. Hanger: http://www.reddit.com/r/coathangersgenral/comments/18s6wp/rugged_jacket_hanger/ 5. Limit Notation at 2: http://www.wolframalpha.com/input/?i=the+limit+of+f%28x%29%3D%28x%5E2-1%29%2F%28x-1%29+as+x+approaches+1+from+the+left 6. Graph of limit DNE: http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html 7. Mean Girls: https://www.youtube.com/watch?v=oDAKKQuBtDo 8. Sin(x) graph: http://www.analyzemath.com/trigonometry/properties.html 9. Hive Statistics: http://www.backyardbeekeepers.com/facts.html 10. Beehive: http://www.umdrightnow.umd.edu/news/nearly-1-3-us-honeybees-lost-winter-2012-13 11. Full Hive: http://www.everythingabout.net/articles/biology/animals/arthropods/insects/bees/cross_section_of_hive.shtml 12. Bee and Flower: http://www.stopsprayingcalifornia.com/Honey_Bees_&_Chemtrails.php 13. Limit at 0 notation: http://tinyurl.com/nzq9dre 14. Limit approaching 0 notation: http://www.wolframalpha.com/input/?i=limit+of+1%2F%282x%29+as+x+approaches+0 15. Positive/Negative: http://www.yolandadeloach.com/master-the-negative/

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The Sky is the Limit  

This is originally designed as an interactive ebook where students can learn the introduction to limits in calculus.

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