## Educators

MM

Problem 1

Describing Notation Write a brief description of the meaning of the notation $\lim _{x \rightarrow 8} f(x)=25$

Amy J.

Problem 2

Limits That Fail to Exist Identify three types of behavior associated with the nonexistence of a limit.
Illustrate each type with a graph of a function.

MM
Meghan M.

Problem 3

Formal Definition of Limit Given the limit

$\lim _{x \rightarrow 2}(2 x+1)=5$
use a sketch to show the meaning of the phrase
$$" 0<|x-2|<0.25 implies |(2 x+1)-5|<0.5"$$

Check back soon!

Problem 4

Functions and Limits Is the limit of $f(x)$ as $x$ approaches calways equal to $f(c) ?$ Why or why not?

MM
Meghan M.

Problem 5

Estimating a Limit Numerically In Exercises
$5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph
the function to confirm your result.

$$\lim _{x \rightarrow 4} \frac{x-4}{x^{2}-5 x+4}$$

Amy J.

Problem 6

Estimating a Limit Numerically In Exercises $5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}$$

Weston B.

Problem 7

Estimating a Limit Numerically In Exercises $5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}$$

Amy J.

Problem 8

Estimating a Limit Numerically In Exercises $5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x}$$

Amy J.

Problem 9

Estimating a Limit Numerically In Exercises $5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\sin x}{x}$$

Amy J.

Problem 10

Estimating a Limit Numerically In Exercises $5-10$ , complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\cos x-1}{x}$$

Weston B.

Problem 11

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$\lim _{x \rightarrow 1} \frac{x-2}{x^{2}+x-6}$

Amy J.

Problem 12

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+9 x+20}$$

Amy J.

Problem 13

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{6}-1}$$

Amy J.

Problem 14

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow-3} \frac{x^{3}+27}{x+3}$$

Check back soon!

Problem 15

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6}$$

Amy J.

Problem 16

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 2} \frac{[x /(x+1)]-(2 / 3)}{x-2}$$

Amy J.

Problem 17

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}$$

Amy J.

Problem 18

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow 0} \frac{\tan x}{\tan 2 x}$$

Amy J.

Problem 19

Limits That Fail to Exist In Exercises 19 and 20 , create a table of values for the function and use the result to explain why the limit does not exist.

$$\lim _{x \rightarrow 0} \frac{2}{x^{3}}$$

Amy J.

Problem 20

In Exercises 19 and 20, create a table of values for the function and use the result to explain why the limit does not exist.

$$\lim _{x \rightarrow 0} \frac{3|x|}{x^{2}}$$

Amy J.

Problem 21

Finding a Limit Graphically In Exercises $21-28$ , use the graph to find the limit (if it exists).
If the limit does not exist, explain why.

$$\lim _{x \rightarrow 3}(4-x)$$

Amy J.

Problem 22

In Exercises21-28, use the graph to find the limit (if it exists).If the limit does not exist, explain why.
$\lim _{x \rightarrow 0} \sec x$

MM
Meghan M.

Problem 23

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$f(x)=\left\{\begin{array}{ll}{4-x,} & {x \neq 2} \\ {0,} & {x=2}\end{array}\right.$$

Amy J.

Problem 24

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 1} f(x)$$

MM
Meghan M.

Problem 25

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$$

Amy J.

Problem 26

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.$$\lim _{x \rightarrow 5} \frac{2}{x-5}$$

MM
Meghan M.

Problem 27

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 0} \cos \frac{1}{x}$$

Amy J.

Problem 28

In Exercises 21-28, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow \pi / 2} \tan x$$

Amy J.

Problem 29

Graphical Reasoning In Exercises 29 and 30 ,use the graph of the function $f$ to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.

(a) $f(1)$
(b) $\lim _{x \rightarrow 1} f(x)$
(c ) $f(4)$
(d) $\lim _{x \rightarrow 4} f(x)$

Amy J.

Problem 30

Graphical Reasoning In Exercises 29 and 30 ,use the graph of the function $f$ to decide whether the value of the given quantity exists. If it does, find it. If not, explain why

(a) $f(-2)$
(b) $\lim _{x \rightarrow-2} f(x)$
(c) $f(0)$
(d) $\lim _{x \rightarrow 0} f(x)$
(e) $f(2)$
(f) $\lim _{x \rightarrow 2} f(x)$
(g) $f(4)$
(h) $\lim _{x \rightarrow 4} f(x)$

MM
Meghan M.

Problem 31

In Exercises 31 and $32,$ sketch the graph of $f$ . Then identify the values of $c$ for which $$\lim _{x \rightarrow e} f(x)$$

$$f(x)=\left\{\begin{array}{ll}{x^{2},} & {x \leq 2} \\ {8-2 x,} & {2 < x < 4} \\ {4,} & {x \geq 4}\end{array}\right.$$

Amy J.

Problem 32

In Exercises 31 and $32,$ sketch the graph of $f$ . Then identify the values of $c$ for which $$\lim _{x \rightarrow e} f(x)$$

$$f(x)=\left\{\begin{array}{ll}{\sin x,} & {x < 0} \\ {1-\cos x,} & {0 \leq x<\pi} \\ {\cos x,} & {x>\pi}\end{array}\right.$$

Amy J.

Problem 33

Sketching a Graph In Exercises 33 and $34,$ sketch a graph of a function $f$ that satisfies the given values. There are many correct answers.)

$f(0)$ is undefined
$$\lim _{x \rightarrow 0} f(x)=4$$
$$f(2)=6$$
$$\lim _{x \rightarrow 2} f(x)=3$$

Amy J.

Problem 34

Sketching a Graph In Exercises 33 and $34,$ sketch a graph of a function $f$ that satisfies the given values. There are many correct answers.)

$$f(-2)=0$$
$$f(2)=0$$
$$\lim _{x \rightarrow-2} f(x)=0$$
$\lim _{x \rightarrow 2} f(x)$ does not exist.

MM
Meghan M.

Problem 35

Finding a $\delta$ for a Given $\varepsilon$ The graph of $f(x)=x+1$ is shown in the figure. Find $\delta$ such that if $0<|x-2|<\delta$ , then $|f(x)-3|<0.4 .$

Amy J.

Problem 36

Finding a $\delta$ for a Given $\varepsilon$ The graph of

$f(x)=\frac{1}{x-1}$

is shown in the figure. Find $\delta$ such that if $0<|x-2|<\delta$ then $|f(x)-1|<0.01$

Amy J.

Problem 37

Finding a $\delta$ for a Given $\varepsilon$ The graph of

$f(x)=2-\frac{1}{x}$

is shown in the figure. Find $\delta$ such that if $0<|x-1|<\delta$ then $|f(x)-1|<0.1$

Amy J.

Problem 38

Finding a $\delta$ for a Given $\varepsilon$ Repeat Exercise 37 for $\varepsilon=0.05,0.01,$ and $0.005 .$ What happens to the value of $\delta$ as the value of $\varepsilon$ gets smaller?

Check back soon!

Problem 39

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$

$$\lim _{x \rightarrow 4}(x+2)$$

Amy J.

Problem 40

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$

$$\lim _{x \rightarrow 6}\left(6-\frac{x}{3}\right)$$

Check back soon!

Problem 41

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$
$$\lim _{x \rightarrow 2}\left(x^{2}-3\right)$$

Amy J.

Problem 42

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$

$$\lim _{x \rightarrow 4}\left(x^{2}+6\right)$$

Check back soon!

Problem 43

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$
$$\lim _{x \rightarrow 4}\left(x^{2}-x\right)$$

Amy J.

Problem 44

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$
$$\lim _{x \rightarrow 3} x^{2}$$

Check back soon!

Problem 45

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .

$$\lim _{x \rightarrow 4}(x+2)$$

Amy J.

Problem 46

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow-2}(4 x+5)$$

MM
Meghan M.

Problem 47

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow-4}\left(\frac{1}{2} x-1\right)$$

Amy J.

Problem 48

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 3}\left(\frac{3}{4} x+1\right)$$

MM
Meghan M.

Problem 49

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 6} 3$$

Amy J.

Problem 50

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 2}(-1)$$

MM
Meghan M.

Problem 51

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 0} \sqrt[3]{x}$$

Amy J.

Problem 52

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 4} \sqrt{x}$$

MM
Meghan M.

Problem 53

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow-5}|x-5|$$

Amy J.

Problem 54

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 3}|x-3|$$

Check back soon!

Problem 55

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow 1}\left(x^{2}+1\right)$$

Check back soon!

Problem 56

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow-4}\left(x^{2}+4 x\right)$$

Check back soon!

Problem 57

Finding a Limit What is the limit of $f(x)=4$ as $x$ apprraches $\pi ?$

Amy J.

Problem 58

Finding a Limit What is the limit of $g(x)=x$ as $x$ approaches $\pi ?$

MM
Meghan M.

Problem 59

Writing In Exercises 59 and $60,$ use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph
about the importance of examining a function analytically as well as graphically.

$$f(x)=\frac{\sqrt{x+5}-3}{x-4}$$

$$\lim _{x \rightarrow 4} f(x)$$

Check back soon!

Problem 60

Writing In Exercises 59 and $60,$ use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph
about the importance of examining a function analytically as well as graphically.

$$\lim _{x \rightarrow 4} f(x)$$f(x)=\frac{x-3}{x^{2}-4 x+3}\lim _{x \rightarrow 3} f(x)$$Check back soon! Problem 61 Modeling Data For a long-distance phone call, a hotel charges \ 9.99 for the first minute and \ 0.79 for each additional minute or fraction thereof. A formula for the cost is given by C(t)=9.99-0.79[1-t], \quad t>0 where t is the time in minutes. (Note: [x]= greatest integer n such that n \leq x . For example, [3.2]=3 and \mathbb{I}-1.6 \mathbb{l}=-2 . ) (a) Evaluate C(10.75) . What does C(10.75) represent? (b) Use a graphing utility to graph the cost function for 0<t \leq 6 . Does the limit of C(t) as t approaches 3 exist? Explain. Check back soon! Problem 62 Modeling Data Repeat Exercise 61 for C(t)=5.79-0.99[1-t], \quad t>0 Check back soon! Problem 63 Finding \delta When using the definition of limit to prove that L is the limit of f(x) as x approaches c, you find the largest satisfactory value of \delta . Why would any smaller positive value of \delta also work? Amy J. Numerade Educator Problem 64 Using the Definition of Limit The definition of limit on page 56 requires that f is a function defined on an open interval containing c , except possibly at c . Why is this requirement necessary? MM Meghan M. Numerade Educator Problem 65 Comparing Functions and Limits Iff(2)=4 can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning. Amy J. Numerade Educator Problem 66 Comparing Functions and Limits If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2) ? Explain your reasoning. MM Meghan M. Numerade Educator Problem 67 Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) The inner circumference of the ring varies between 5.5 centimeters and 6.5 centimeters. How does the radius \quad vary? (c) Use the \varepsilon-\delta definition of limit to describe this situation. Identify \varepsilon and \delta . Check back soon! Problem 68 A sporting goods manufacturer designs a golf ball having a : volume of 2.48 cubic inches. (a) What is the radius of the golf ball? (b) The volume of the golf ball varies between 2.45 cubic inches. How does the radius vary? (c) Use the \varepsilon-\delta definition of limit to describe this situation. Identify \varepsilon and \delta . Check back soon! Problem 69 Estimating a Limit Consider the function f(x)=(1+x)^{1 / x} Estimate$$\lim _{x \rightarrow 0}(1+x)^{1 / x}$$by evaluating f at x -values near 0 . Sketch the graph of f Check back soon! Problem 70 by evaluating f at x -values near 0 . Sketch the graph of f f(x)=\frac{|x+1|-|x-1|}{x} Estimate \lim _{x \rightarrow 0} \frac{|x+1|-|x-1|}{x} by evaluating f at x -values near 0 . Sketch the graph of f Check back soon! Problem 71 Graphical Reasoning The statement$$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4$$means that for each \varepsilon>0 there corresponds a \delta>0 such that if 0<|x-2|<\delta, then$$\left|\frac{x^{2}-4}{x-2}-4\right|<\varepsilon$$If \varepsilon=0.001, then$$\left|\frac{x^{2}-4}{x-2}-4\right|<0.001$$Use a graphing utility to graph each side of this inequality. Use the zoom feature to find an interval (2-\delta, 2+\delta) such that the inequality is true. Check back soon! Problem 72 HOW DO YOU SEE IT? Use the graph of f to identify the values of c for which lim f(x) exists. Check back soon! Problem 73 True or False? In Exercises 73-76, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is undefined at x=c , then the limit of f(x) as x approaches c does not exist. Amy J. Numerade Educator Problem 74 True or False? In Exercises 73-76 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)<0.001 Check back soon! Problem 75 True or False? In Exercises 73-76 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If$$f(c)=L,$then$\lim _{\mathrm{lim}} f(x)=L$$Amy J. Numerade Educator Problem 76 True or False? In Exercises 73-76 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If$$\lim _{x \rightarrow c} f(x)=L,$then$f(c)=L$$MM Meghan M. Numerade Educator Problem 77 Determining a Limit In Exercises 77 and 78, consider the function f(x)=\sqrt{x} . Is$$\lim _{x \rightarrow 0.25} \sqrt{x}=0.5$$a true statement? Explain. Amy J. Numerade Educator Problem 78 Determining a Limit In Exercises 77 and 78, consider the function f(x)=\sqrt{x} . Is$$\lim _{x \rightarrow 0} \sqrt{x}=0$$a true statement? Explain. MM Meghan M. Numerade Educator Problem 79 Evaluating Limits Use a graphing utility to evaluate$$\lim _{x \rightarrow 0} \frac{\sin n x}{x}$$for several values of n . What do you notice? Amy J. Numerade Educator Problem 80 Evaluating Limits Use a graphing utility to evaluate$$\lim _{x \rightarrow 0} \frac{\tan n x}{x}$$for several values of n . What do you notice? Check back soon! Problem 81 Proof Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique. [Hint: Let$$\lim _{x \rightarrow c} f(x)=L_{1}$$and$$\lim _{x \rightarrow c} f(x)=L_{2}$$and prove that L_{1}=L_{2} . ] Check back soon! Problem 82 Proof Consider the line f(x)=m x+b, where m \neq 0 . Use the \varepsilon -\delta definition of limit to prove that$$\lim _{x \rightarrow c} f(x)=m c+b$$MM Meghan M. Numerade Educator Problem 83 Proof Prove that$$\lim _{x \rightarrow c} f(x)=L$$is equivalent to$$\lim _{x \rightarrow c}[f(x)-L]=0$$Amy J. Numerade Educator Problem 84 Proof (a) Given that$$\lim _{x \rightarrow 0}(3 x+1)(3 x-1) x^{2}+0.01=0.01$$prove that there exists an open interval (a, b) containing 0 such that (3 x+1)(3 x-1) x^{2}+0.01>0 for all x \neq 0 in (a, b) . (b) Given that$$\lim _{x \rightarrow c} g(x)=L, where $L>0,$ prove that there
exists an open interval $(a, b)$ containing $c$ such that
$g(x)>0$ for all $x \neq c$ in $(a, b) .$

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Problem 85

Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in a region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area?

Weston B.