A Graphical Approach to Precalculus with Limits

John Hornsby, Margaret L. Lial, Gary Rockswold

Chapter 13

Limits, Derivatives, and Definite Integrals - all with Video Answers

Educators

Section 1

An Introduction to Limits

Problem 1

Tell whether each statement is true or false.
If a function $f$ is defined at $x=a,$ then $\lim f(x)$ is always equal to $f(a)$

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Problem 2

Tell whether each statement is true or false.
If $\lim f(x)$ does not exist, then $f(x)$ necessarily approaches one value as $x$ approaches $a$ from the left and a different value as $x$ approaches $a$ from the right.

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Problem 3

Tell whether each statement is true or false.
If $\lim _{x \rightarrow 1} f(x)=5,$ then 1 must be in the domain of $f(x)$

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Problem 4

Tell whether each statement is true or false.
If $\lim _{x \rightarrow 1} f(x)=5,$ then 1 must be in the domain of $f(x)$

Darshan Maheshwari

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Problem 5

Tell whether each statement is true or false.
If $\lim _{x \rightarrow a} f(x)=-5,$ then $f(x)$ is between $-5.001$ and $-4.999$ for some value of $x$ near $a$

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Problem 6

Tell whether each statement is true or false.
If $\lim _{x \rightarrow a} f(x)=b,$ then $|f(x)-b|<0.0001$ for some
value of $x$ near $a$

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Problem 7

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 8

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 9

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 10

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 11

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 12

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 13

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 14

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 15

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 16

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 17

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

Darshan Maheshwari

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Problem 18

Decide from the graph whether each limit exists. If a limit exists, find its value. Do not use a calculator.

Darshan Maheshwari

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Problem 19

Use the table of values to predict $\lim _{x \rightarrow 1} f(x)$
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\
\hline f(x) & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1
\end{array}$$

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Problem 20

Use the table of values to predict $\lim _{x \rightarrow 2} f(x)$
$$\begin{array}{|c|r|r|r|r|c|c|}
\hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\
\hline f(x) & -1.3 & -1.05 & -1.002 & -0.997 & -0.993 & -0.985
\end{array}$$

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Problem 21

Complete each table and use the results to predict the indicated limit, if it exists.
If $f(x)=2 x^{2}-4 x+3,$ find $\lim _{x \rightarrow 1} f(x)$
$$\begin{array}{|c|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & & & 1.000002 & 1.000002 & & \end{array}$$

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Problem 22

Complete each table and use the results to predict the indicated limit, if it exists.
$$\text { If } f(x)=\frac{2 x^{3}+3 x^{2}-4 x-5}{x+1}, \text { find } \lim _{x \rightarrow-1} f(x)$$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -1.1 & -1.01 & -1.001 & -0.999 & -0.99 & -0.9 \\ \hline f(x) & -3.68 & & & & & -4.28 \end{array}$$

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Problem 23

Complete each table and use the results to predict the indicated limit, if it exists.
If $f(x)=\frac{x^{3}-2 x-4}{x-2},$ find $\lim _{x \rightarrow 2} f(x)$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & 9.9401 & & & 10.0601 & \end{array}$$

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Problem 24

Complete each table and use the results to predict the indicated limit, if it exists.
$$\text { If } f(x)=\frac{\sqrt{x}-3}{x-3}, \text { find } \lim _{x \rightarrow 3} f(x)$$
$$\begin{array}{|c|l|l|l|l|l|l|} \hline x & 2.9 & 2.99 & 2.999 & 3.001 & 3.01 & 3.1 \\ \hline f(x) & & & & & & \end{array}$$

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Problem 25

Complete each table and use the results to predict the indicated limit, if it exists.
If $f(x)=\frac{\sqrt{x}-2}{x-1},$ find $\lim _{x \rightarrow 1} f(x)$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & & & & & & \end{array}$$

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Problem 26

Complete each table and use the results to predict the indicated limit, if it exists.
$$\text { If } f(x)=\frac{x^{3}+3 x^{2}+x+3}{x+3}, \text { find } \lim _{x \rightarrow-3} f(x)$$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -3.1 & -3.01 & -3.001 & -2.999 & -2.99 & -2.9 \\ \hline f(x) & & & & & & \end{array}$$

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Problem 27

Complete each table and use the results to predict the indicated limit, if it exists.
If $f(x)=\frac{\sin 2 x}{x},$ find $\lim _{x \rightarrow 0} f(x)$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \end{array}$$

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Problem 28

Complete each table and use the results to predict the indicated limit, if it exists.
$$\text { If } f(x)=\frac{\sin 5 x}{2 x}, \text { find } \lim _{x \rightarrow 0} f(x)$$
$$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \end{array}$$

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Problem 29

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 5}|2 x-4|$

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Problem 30

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-1} \sqrt{5+3 x}$

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Problem 31

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 5} \frac{x^{2}-3 x-10}{x-5}$

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Problem 32

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-4} \frac{x^{2}+5 x+4}{x+4}$

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Problem 33

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-2} \frac{x^{2}+2}{x+2}$

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Problem 34

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 3} \frac{x+1}{(x-3)^{2}}$

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Problem 35

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2}$

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Problem 36

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 2} \frac{x^{2}-3 x+2}{x-2}$

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Problem 37

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-3} \frac{2 x^{2}+5 x-3}{x+3}$

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Problem 38

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}$

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Problem 39

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1}$

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Problem 40

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-6 x+9}$

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Problem 41

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-1} \frac{x+1}{x^{2}+2 x+1}$

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Problem 42

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} \frac{e^{x-1}-x}{x-1}$

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Problem 43

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 3} f(x),$ where
$f(x)=\left\{\begin{array}{ll}x+7 & \text { if } x \leq 3 \\ 5 x-5 & \text { if } x>3\end{array}\right.$

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Problem 44

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow-1} f(x),$ where $f(x)=\left\{\begin{array}{ll}\sqrt{3-x} & \text { if } x<-1 \\ \frac{4}{1-x} & \text { if } x>-1\end{array}\right.$

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Problem 45

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 2} f(x),$ where $f(x)=\left\{\begin{array}{ll}x^{2}-3 & \text { if } x<2 \\ 5-x^{2} & \text { if } x>2\end{array}\right.$

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Problem 46

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} f(x),$ where $f(x)=\left\{\begin{array}{ll}3 x-5 & \text { if } x \leq 1 \\ 6-2 x & \text { if } x>1\end{array}\right.$

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Problem 47

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} f(x),$ where $f(x)=\left\{\begin{array}{ll}e^{x} & \text { if } x \leq 1 \\ \sqrt{x} & \text { if } x>1\end{array}\right.$

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Problem 48

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$

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Problem 49

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{x^{3}}{x-\sin x}$

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Problem 50

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{\sin x}{\sin 2 x}$

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Problem 50

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{\sin x}{\sin 2 x}$

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Problem 51

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{\cos x-1}{x}$

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Problem 52

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow \pi} \frac{\tan ^{2} x}{1+\sec x}$

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Problem 53

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}-1}$

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Problem 54

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} \frac{\ln x}{x-1}$

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Problem 55

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 1} \frac{\ln x^{2}}{\ln x}$

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Problem 56

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{e^{-x}-1}{x}$

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Problem 57

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0}(x \sin x)$

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Problem 58

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0}(x \ln |x|)$

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Problem 59

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \tan \frac{1}{x}$

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Problem 60

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \cos \frac{1}{x}$

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Problem 61

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \cos \frac{1}{x}$

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Problem 62

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{2 x}{\tan x}$

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Problem 63

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0}(x \csc x)$

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Problem 64

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value.
$\lim _{x \rightarrow 0} \frac{e^{x}-1}{2 x}$

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