Chapter 1

Limits and Their Properties

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.
Numerade Educator

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.
Numerade Educator

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
$$" 0<|x-2|<0.25$ implies $|(2 x+1)-5|<0.5"$$

Katelyn C.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.

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

Katelyn C.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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?

Katelyn C.
Numerade Educator

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.
Numerade Educator

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 $(b) \varepsilon=0.005 .$

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

Katelyn C.
Numerade Educator

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.
Numerade Educator

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 $(b) \varepsilon=0.005 .$

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

Katelyn C.
Numerade Educator

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.
Numerade Educator

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 $(b) \varepsilon=0.005 .$
$$\lim _{x \rightarrow 3} x^{2}$$

Katelyn C.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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.
Numerade Educator

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 $L$ .
$$\lim _{x \rightarrow 1}\left(x^{2}+1\right)$$

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

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Problem 57

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

Amy J.
Numerade Educator

Problem 58

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

MM
Meghan M.
Numerade Educator

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.

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

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

Katelyn C.
Numerade Educator

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}$$

$$\lim _{x \rightarrow 3} f(x)$$

Katelyn C.
Numerade Educator

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.

Katelyn C.
Numerade Educator

Problem 62

Modeling Data Repeat Exercise 61 for

$C(t)=5.79-0.99[1-t], \quad t>0$

Katelyn C.
Numerade Educator

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 If$f(2)=4$ 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 $\delta .$

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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 $\delta .$

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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 $f$

Katelyn C.
Numerade Educator

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 $f$

Katelyn C.
Numerade Educator

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.

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Problem 72

HOW DO YOU SEE IT? Use the graph of $f$ to identify the values of $c$ for which lim $f(x)$ exists.

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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 $f(k)<0.001$

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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?

Katelyn C.
Numerade Educator

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 $L_{1}=L_{2} . ]$

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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.
Numerade Educator

Problem 86

A right circular cone has base of radius 1 and height 3.A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?

Weston B.
Numerade Educator