Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

Chapter 3

The Derivative - all with Video Answers

Educators

+ 2 more educators

Section 1

Limits

Problem 1

In Exercises 1-4, choose the best answer for each limit.
$$If \lim _{x \rightarrow 2} f(x)=5 and \lim _{x \rightarrow 2^{-}} f(x)=6,$ then \lim _{x \rightarrow 2} f(x)$$
$$\begin{array}{ll}{\text { (a) is } 5 .} & {\text { (b) is } 6} \\ {\text { (c) does not exist. }} & {\text { (d) is infinite. }}\end{array}$$

Todd Vawdrey

Todd Vawdrey

Numerade Educator

Problem 2

In Exercises 1-4, choose the best answer for each limit.
$$If \lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=-1, but f(2)=1, then \lim _{x \rightarrow 2} f(x)$$
$$\begin{array}{ll}{\text { (a) is }-1 .} & {\text { (b) does not exist. }} \\ {\text { (c) is infinite. }} & {\text { (d) is } 1 .}\end{array}$$

Helen Telila

Numerade Educator

Problem 3

In Exercises 1-4, choose the best answer for each limit.
$$If \lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)=5, but f(4) does not exist, then
\lim _{x \rightarrow 4^{-}} f(x)$$
$$\begin{array}{ll}{\text { (a) is } 5} & {\text { (b) is }-\infty} \\ {\text { (c) is }+\infty} & {\text { (d) does not exist }}\end{array}$$

Todd Vawdrey

Numerade Educator

Problem 4

In Exercises 1-4, choose the best answer for each limit.
$$If \lim _{x \rightarrow 1} f(x)=-\infty and \lim _{x \rightarrow 1^{+}} f(x)=-\infty, then \lim _{x \rightarrow 1} f(x)$$
$$\begin{array}{ll}{\text { (a) is } \infty .} & {\text { (b) is }-\infty.} \\ {\text { (c) does not exist. }} & {\text { (d) is } 1 \text { . }}\end{array}$$

Helen Telila

Numerade Educator

Problem 5

Decide whether each limit exists. If a limit exists, estimate its
value.
$${ (a) }\lim _{x \rightarrow 3} f(x) \quad \text { (b) } \lim _{x \rightarrow 0} f(x)
$$

Todd Vawdrey

Numerade Educator

Problem 6

Decide whether each limit exists. If a limit exists, estimate its
value.
$${ (a) }\lim _{x \rightarrow 2} F(x) \quad \text { (b) } \lim _{x \rightarrow-1} F(x)
$$

Helen Telila

Numerade Educator

Problem 7

Decide whether each limit exists. If a limit exists, estimate its value.
$${ (a) }\lim _{x \rightarrow 0} f(x) \quad \text { (b) } \lim _{x \rightarrow 2} f(x)
$$

Todd Vawdrey

Numerade Educator

Problem 8

Decide whether each limit exists. If a limit exists, estimate its value.
$${ (a) } \lim _{x \rightarrow 3} g(x) \quad \text { (b) } \lim _{x \rightarrow 5} g(x)$$

Helen Telila

Numerade Educator

Problem 9

In Exercises 9 and $10,$ use the graph to find (i) $\lim _{x \rightarrow a^{-}} f(x)$
(ii) $\lim f(x),($ iii) $\lim f(x)$ , and $($ iv $) f(a)$ if it exists.
$${ (a) } a=-2 \quad \text { (b) } a=-1$$

Julian Wong

Numerade Educator

Problem 10

In Exercises 9 and $10,$ use the graph to find (i) $\lim _{x \rightarrow a^{-}} f(x)$
(ii) $\lim f(x),($ iii) $\lim f(x)$ , and $($ iv $) f(a)$ if it exists.
$${ (a) } a=1 \quad \text { (b) } a=2$$

Helen Telila

Numerade Educator

Problem 11

Decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} f(x)$$

Julian Wong

Numerade Educator

Problem 12

Decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow-\infty} g(x)$$

Helen Telila

Numerade Educator

Problem 13

Explain why lim $F(x)$ in Exercise 6 exists, but $\lim _{x \rightarrow-2} f(x)$ in
Exercise 9 does not.

Vikash Ranjan

Numerade Educator

Problem 14

In Exercise $10,$ why does $\lim _{x \rightarrow 1} f(x)=1,$ even though $f(1)=2 ?$

Helen Telila

Numerade Educator

Problem 15

Use the table of values to estimate lim $f(x)$
$$\begin{array}{r|c|cccccccc}{x} & {0.9} & {0.99} & {0.999} & {0.9999} & {1.0001} & {1.001} & {1.01} & {1.1} \\ {f(x)} & {3.9} & {3.99} & {3.999} & {3.9999} & {4.0001} & {4.001} & {4.01} & {4.1}\end{array}$$

Julian Wong

Numerade Educator

Problem 16

Complete the tables and use the results to find the indicated
limits.
If $f(x)=2 x^{2}-4 x+7,$ find $\lim _{x \rightarrow 1} f(x)$

Helen Telila

Numerade Educator

Problem 17

$$If k(x)=\frac{x^{3}-125}{x-5}, find \lim _{x \rightarrow 5} k(x)$$
$$\begin{array}{l}{x \quad 4.9 \quad 4.99 \quad 4.999 \quad 5.001 \quad 5.01 \quad 5.1} \\ {k(x)}\end{array}$$

Julian Wong

Numerade Educator

Problem 18

$$If f(x)=\frac{x^{3}-729}{x-9}, find \lim _{x \rightarrow 9} f(x)$$
$$\begin{array}{l}{x \quad 8.9 \quad 8.99 \quad 8.999 \quad 9.001 \quad 9.01 \quad 9.1} \\ {f(x)}\end{array}$$

Helen Telila

Numerade Educator

Problem 19

$$h(x)=\frac{\sqrt{x}+2}{x-2}, \text { then find } \lim _{x \rightarrow 2} h(x)$$
$$\begin{array}{l}{x \quad 1.9 \quad 1.99 \quad 1.999 \quad 2.001 \quad 2.01 \quad 2.1} \\ {h(x)}\end{array}$$

Julian Wong

Numerade Educator

Problem 20

$$If h(x)=\frac{\sqrt{x}+2}{x-8}, find \lim _{x \rightarrow 8} h(x)$$
$$\begin{array}{l}{x \quad 7.9 \quad 7.99 \quad 7.999 \quad 8.001 \quad 8.01 \quad 8.1} \\ {h(x)}\end{array}$$

Helen Telila

Numerade Educator

Problem 21

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find
each limit.
$$\lim _{x \rightarrow 4}[f(x)-g(x)]$$

Todd Vawdrey

Numerade Educator

Problem 22

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4}[g(x) \cdot f(x)]$$

Helen Telila

Numerade Educator

Problem 23

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 24

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \log _{3} f(x)$$

Helen Telila

Numerade Educator

Problem 25

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \sqrt{f(x)}$$

Julian Wong

Numerade Educator

Problem 26

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \sqrt[3]{g(x)}$$

Helen Telila

Numerade Educator

Problem 27

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} 2^{f(x)}$$

Julian Wong

Numerade Educator

Problem 28

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4}[1+f(x)]^{2}$$

Helen Telila

Numerade Educator

Problem 29

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \frac{[f(x)+g(x)]}{2 g(x)}$$

Julian Wong

Numerade Educator

Problem 30

Let $\lim _{x \rightarrow 4} f(x)=9$ and $\lim _{x \rightarrow 4} g(x)=27 .$ Use the limit rules to find each limit.
$$\lim _{x \rightarrow 4} \frac{5 g(x)+2}{1-f(x)}$$

Helen Telila

Numerade Educator

Problem 31

Use the properties of limits to help decide whether each limit
exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}$$

Nicholas Carlson

Nicholas Carlson

Numerade Educator

Problem 32

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 10} \frac{x^{2}-100}{x-10}$$

Helen Telila

Numerade Educator

Problem 33

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 4} \frac{x^{2}+2 x-24}{x^{2}-16}$$

Julian Wong

Numerade Educator

Problem 34

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 3} \frac{x^{2}+4 x-21}{x^{2}-9}$$

Helen Telila

Numerade Educator

Problem 35

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 8} \frac{x^{2}-2 x-48}{x-8}$$

Todd Vawdrey

Numerade Educator

Problem 36

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 9} \frac{x^{2}-3 x-54}{x-9}$$

Helen Telila

Numerade Educator

Problem 37

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 0} \frac{1 /(x+3)-1 / 3}{x}$$

Julian Wong

Numerade Educator

Problem 38

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 0} \frac{-1 /(x+2)+1 / 2}{x}$$

Helen Telila

Numerade Educator

Problem 39

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 169} \frac{\sqrt{x}-13}{x-169}$$

Julian Wong

Numerade Educator

Problem 40

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}$$

Helen Telila

Numerade Educator

Problem 41

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h}$$

Julian Wong

Numerade Educator

Problem 42

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{h \rightarrow 0} \frac{(x+h)^{3}-x^{3}}{h}$$

Helen Telila

Numerade Educator

Problem 43

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{7 x}{8 x-5}$$

Julian Wong

Numerade Educator

Problem 44

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow-\infty} \frac{8 x+2}{4 x-5}$$

Helen Telila

Numerade Educator

Problem 45

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow-\infty} \frac{9 x^{2}+2 x}{8 x^{2}-6 x+1}$$

Julian Wong

Numerade Educator

Problem 46

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x-5}{3 x^{2}+2}$$

Helen Telila

Numerade Educator

Problem 47

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{7 x^{3}+8 x-8}{5 x^{4}-9 x^{3}-6}$$

Julian Wong

Numerade Educator

Problem 48

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{2 x^{2}-1}{3 x^{4}+2}$$

Helen Telila

Numerade Educator

Problem 49

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{2 x^{3}-5 x-6}{6 x^{2}-2 x-3}$$

Julian Wong

Numerade Educator

Problem 50

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{x^{4}-x^{3}-3 x}{7 x^{2}+9}$$

Helen Telila

Numerade Educator

Problem 51

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{5 x^{2}-4 x^{4}}{9 x^{2}-9 x-6}$$

Julian Wong

Numerade Educator

Problem 52

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$\lim _{x \rightarrow \infty} \frac{-5 x^{3}-4 x^{2}+8}{6 x^{2}+3 x+2}$$

Maria Dearborn

Numerade Educator

Problem 53

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$Letf(x)=\left\{\begin{array}{ll}{x^{3}+2} & {\text { if } x \neq-1} \\ {5} & {\text { if } x=-1}\end{array}\right. \quad \text { Find } \lim _{x \rightarrow-1} f(x)$$

Julian Wong

Numerade Educator

Problem 54

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$Let g(x)=\left\{\begin{array}{ll}{0} & {\text { if } x=-2} \\ {\frac{1}{2} x^{2}-3} & {\text { if } x \neq-2}\end{array}\right. \quad Find \lim _{x \rightarrow-2} g(x)$$

Helen Telila

Numerade Educator

Problem 55

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$Letf(x)=\left\{\begin{array}{ll}{x-1} & {\text { if } x<3} \\ {2} & {\text { if } 3 \leq x \leq 5} \\ {x+3} & {\text { if } x>5}\end{array}\right.$$
$${ (b) Find }\lim _{x \rightarrow 3} f(x) . \quad \text { (b) Find } \lim _{x \rightarrow 5} f(x)$$

Julian Wong

Numerade Educator

Problem 56

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$Letg(x)=\left\{\begin{array}{ll}{5} & {\text { if } x<0} \\ {x^{2}-2} & {\text { if } 0 \leq x \leq 3} \\ {7} & {\text { if } x>3}\end{array}\right.$$
$${ (a) Find }\lim _{x \rightarrow 0} g(x) . \quad \text { (b) Find } \lim _{x \rightarrow 3} g(x)$$

Helen Telila

Numerade Educator

Problem 57

Does a value of $k$ exist such that the following limit exists?
$$\lim _{x \rightarrow 2} \frac{3 x^{2}+k x-2}{x^{2}-3 x+2}$$
If so, find the value of $k$ and the corresponding limit. If not,
explain why not.

Julian Wong

Numerade Educator

Problem 58

Repeat the instructions of Exercise 57 for the following limit.
$$\lim _{x \rightarrow 3} \frac{2 x^{2}+k x-9}{x^{2}-4 x+3}$$

Helen Telila

Numerade Educator

Problem 59

In Exercises 59-62, calculate the limit in the specified exercise,
using a table such as in Exercises 15-20. Verify your answer
by using a graphing calculator to zoom in on the point on the
graph.
Exercise 31

Julian Wong

Numerade Educator

Problem 60

Exercise 32

Helen Telila

Numerade Educator

Problem 61

Exercise 33

Julian Wong

Numerade Educator

Problem 62

Exercise 34

Helen Telila

Numerade Educator

Problem 63

$$\begin{array}{l}{\text { Let } F(x)=\frac{3 x}{(x+2)^{3.}}} \\ {\text { (a) Find } \lim _{x \rightarrow-2} F(x.)} \\ {\text { (b) Find the vertical asymptote of the graph of } F(x).} \\ {\text { (c) Compare your answers for parts (a) and (b). What can you }} \\ {\text { conclude? }}\end{array}$$

Julian Wong

Numerade Educator

Problem 64

$$\begin{array}{l}{\text { Let } G(x)=\frac{-6}{(x-4)^{2.}}} \\ {\text { (a) Find } \lim _{x \rightarrow 4} G(x)} \\ {\text { (b) Find the vertical asymptote of the graph of } G(x).} \\ {\text { (c) Compare your answers for parts (a) and (b). Are they related? }} \\ {\text { How? }}\end{array}$$

Helen Telila

Numerade Educator

Problem 65

Describe how the behavior of the graph in Figure 10 near x = 1
can be predicted by the simplified expression for the function
$y=1 /(x-1).$

Julian Wong

Numerade Educator

Problem 66

A friend who is confused about limits wonders why you investi-
gate the value of a function closer and closer to a point, instead

of just finding the value of a function at the point. How would
you respond?

Helen Telila

Numerade Educator

Problem 67

$$\begin{array}{l}{\text { Use a graph of } f(x)=e^{x} \text { to answer the following questions. }} \\ {\text { (a) Find } \lim _{x \rightarrow-\infty} e^{x} \text { . }} \\ {\text { (b) Where does the function } e^{x} \text { have a horizontal asymptote? }}\end{array}$$

Julian Wong

Numerade Educator

Problem 68

Use a graphing calculator to answer the following questions.
$$\begin{array}{l}{\text { (a) From a graph of } y=x e^{-x}, \text { what do you think is the value }} \\ {\text { of lim } x e^{-x} ? \text { Support this by evaluating the function for }} \\ {\text { several large values of } x \text { . }}\end{array}$$
(b) Repeat part (a), this time using the graph of $y=x^{2} e^{-x}$
$$\begin{array}{l}{\text { (c) Based on your results from parts (a) and (b), what do }} \\ {\text { you think is the value of } \lim _{x \rightarrow \infty} x^{n} e^{-x} \text { , where } n \text { is a positive }} \\ {\text { integer? Support this by experimenting with other positive }} \\ {\text { integers } n .}\end{array}
$$

Todd Vawdrey

Numerade Educator

Problem 69

Use a graph of $f(x)=\ln x$ to answer the following questions.
(a) Find $\lim _{x \rightarrow 0^{+}} \ln x$
(b) Where does the function ln $x$ have a vertical asymptote?

Julian Wong

Numerade Educator

Problem 70

Use a graphing calculator to answer the following questions.
$$\begin{array}{l}{\text { (a) From a graph of } y=x \ln x, \text { what do you think is the value }} \\ {\text { of } \lim _{x \rightarrow 0^{+}} x \ln x ? \text { Support this by evaluating the function for }} \\ {\text { several small yalues of } x \text { . }}\end{array}$$
$$\begin{array}{l}{\text { (b) Repeat part (a), this time using the graph of } y=x(\ln x)^{2} \text { . }} \\ {\text { (c) Based on your results from parts (a) and (b), what do you think }} \\ {\text { is the value of } \lim _{x \rightarrow 0^{+}} x(\ln x)^{n}, \text { where } n \text { is a positive integer? }} \\ {\text { Support this by experimenting with other positive integers } n .}\end{array}$$

Todd Vawdrey

Numerade Educator

Problem 71

Explain in your own words why the rules for limits at infinity
should be true.

Julian Wong

Numerade Educator

Problem 72

Explain in your own words what Rule 4 for limits means.

Todd Vawdrey

Numerade Educator

Problem 73

Find each of the following limits (a) by investigating values of the function near the $x$ -value where the limit is taken, and (b) using a graphing calculator to view the function near that value of $x .$
$$\lim _{x \rightarrow 1} \frac{x^{4}+4 x^{3}-9 x^{2}+7 x-3}{x-1}$$

Julian Wong

Numerade Educator

Problem 74

Find each of the following limits (a) by investigating values of the function near the $x$ -value where the limit is taken, and (b) using a graphing calculator to view the function near that value of $x .$
$$\lim _{x \rightarrow 2} \frac{x^{4}+x-18}{x^{2}-4}$$

Nick Johnson

Numerade Educator

Problem 75

Find each of the following limits (a) by investigating values of the function near the $x$ -value where the limit is taken, and (b) using a graphing calculator to view the function near that value of $x .$
$$\lim _{x \rightarrow-1} \frac{x^{1 / 3}+1}{x+1}$$

Julian Wong

Numerade Educator

Problem 76

Find each of the following limits (a) by investigating values of the function near the $x$ -value where the limit is taken, and (b) using a graphing calculator to view the function near that value of $x .$
$$\lim _{x \rightarrow 4} \frac{x^{3 / 2}-8}{x+x^{1 / 2}-6}$$

Nick Johnson

Numerade Educator

Problem 77

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}$$

Julian Wong

Numerade Educator

Problem 78

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{2}+5}}{2 x}$$

Julian Wong

Numerade Educator

Problem 79

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow-\infty} \frac{\sqrt{36 x^{2}+2 x+7}}{3 x}$$

Julian Wong

Numerade Educator

Problem 80

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow \infty} \frac{\sqrt{36 x^{2}+2 x+7}}{3 x}$$

Julian Wong

Numerade Educator

Problem 81

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow \infty} \frac{\left(1+5 x^{1 / 3}+2 x^{5 / 3}\right)^{3}}{x^{5}}$$

Julian Wong

Numerade Educator

Problem 82

Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for $f(x).$
$$\lim _{x \rightarrow-\infty} \frac{\left(1+5 x^{1 / 3}+2 x^{5 / 3}\right)^{3}}{x^{5}}$$

Julian Wong

Numerade Educator

Problem 83

Explain why the following rules can be used to find
$\lim _{x \rightarrow \infty}[p(x) / q(x)] :$
$$\begin{array}{l}{\text { (a) If the degree of } p(x) \text { is less than the degree of } q(x), \text { the }} \\ {\text { limit is } 0 .} \\ {\text { (b) If the degree of } p(x) \text { is equal to the degree of } q(x), \text { the }} \\ {\text { limit is } A / B, \text { where } A \text { and } B \text { are the leading coefficients of }} \\ {p(x) \text { and } q(x), \text { respectively. }} \\ {\text { (c) If the degree of } p(x) \text { is greater than the degree of } q(x), \text { the }} \\ {\text { limit is } \infty \text { or }-\infty .}\end{array}$$

Julian Wong

Numerade Educator

Problem 84

APPLY IT Consumer Demand When the price of an
essential commodity (such as gasoline) rises rapidly, consump-
tion drops slowly at first. If the price continues to rise, however,
a "tipping" point may be reached, at which consumption takes
a sudden substantial drop. Suppose the accompanying graph
shows the consumption of gasoline, $G(t),$ in millions of gal-
lons, in a certain area. We assume that the price is rising rapidly.
Here $t$ is time in months after the price began rising. Use the
graph to find the following.
$${ (a) }\lim _{t \rightarrow 12} G(t) \quad \text { (b) } \lim _{t \rightarrow 16} G(t)$$(c) $G(16) \quad$ (d) The tipping point (in months)

Linh Vu

Numerade Educator

Problem 85

Sales Tax Officials in California tend to raise the sales tax
in years in which the state faces a budget deficit and then cut
the tax when the state has a surplus. The graph below shows the California state sales tax in recent years. Let $T(x)$ represent the
sales tax per dollar spent in year $x$ . Find the following. Source:
California State.
$$\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 94} T(x)} & {\text { (b) } \lim _{x \rightarrow 13^{-}} T(x)} \\ {\text { (c) } \lim _{x \rightarrow 13^{*}} T(x)} & {\text { (d) } \lim _{x \rightarrow 13} T(x)} \\ {\text { (e) } T(13)}\end{array}$$$

Julian Wong

Numerade Educator

Problem 86

Postage The graph below shows how the postage required to
mail a letter in the United States has changed in recent years.
Let $C(t)$ be the cost to mail a letter in the year $t$ . Find the fol-
lowing. Source: United States Postal Service.
$${ (a) }\lim _{t \rightarrow 2014} C(t) \quad \text { (b) } \lim _{t \rightarrow 2014^{+}} C(t)$$
$${ (c) }\lim _{t \rightarrow 2014} C(t) \quad \text { (d) } \ C(2014)$$$

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 87

Average Cost The cost (in dollars) for manufacturing a par-
ticular toy is
$$C(x)=25,000+9 x,$$
where $x$ is the number of toys produced. Recall from the
previous chapter that the average cost per toy, denoted by
$\overline{C}(x),$ is found by dividing $C(x)$ by $x .$ Find and interpret
$\lim _{x \rightarrow \infty} \overline{C}(x).$

Julian Wong

Numerade Educator

Problem 88

Average cost In Chapter $1,$ we saw that the cost to fly
$x$ miles on American Airlines could be approximated by the
equation
$$C(x)=0.0417 x+167.55$$
Recall from the previous chapter that the average cost per mile,
denoted by $\overline{C}(x),$ is found by dividing $C(x)$ by $x$ . Find and
interpret $\lim _{x \rightarrow \infty} \widehat{C}(x) .$ Source: American Airlines.

Linh Vu

Numerade Educator

Problem 89

Employee Productivity A company training program has
determined that, on the average, a new employee produces $P(s)$
items per day after s days of on-the-job training where
$P(s)=\frac{56 s}{s+6.}$
Find and interpret $\lim _{s \rightarrow \infty} P(s)$

Julian Wong

Numerade Educator

Problem 90

Preferred Stock In business finance, an annuity is a series of
equal payments received at equal intervals for a finite period
of time. The present value of an $n$ -period annuity takes the
form.
$$P=R\left[\frac{1-(1+i)^{-n}}{i}\right]$$
where $R$ is the amount of the periodic payment and $i$ is the fixed
interest rate per period. Many corporations raise money by
issuing preferred stock. Holders of the preferred stock, calleda perpetuity, receive payments that take the form of an annu-
ity in that the amount of the payment never changes. However,
normally the payments for preferred stock do not end but theo-
retically continue forever. Find the limit of this present value
equation as $n$ approaches infinity to derive a formula for the
present value of a share of preferred stock paying a periodic
dividend $R .$ Source: Robert $D .$ Campbell.

Linh Vu

Numerade Educator

Problem 91

Growing Annuities For some annuities encountered in busi-
ness finance, called growing annuities, the amount of the peri-
odic payment is not constant but grows at a constant periodic
rate. Leases with escalation clauses can be examples of growing
annuities. The present value of a growing annuity takes the form
$$P=\frac{R}{i-g}\left[1-\left(\frac{1+g}{1+i}\right)^{n}\right]$$
where
$$\begin{aligned} R &=\text { amount of the next annuity payment, } \\ g &=\text { expected constant annuity growth rate, } \\ i &=\text { required periodic return at the time the annuity is } \\ & \text { evaluated, } \\ n &=\text { number of periodic payments. } \end{aligned}$$
A corporation's common stock may be thought of as a claim on
a growing annuity where the annuity is the company's annual
dividend. However, in the case of common stock, these payments
have no contractual end but theoretically continue forever. Com-
pute the limit of the expression above as $n$ approaches infinity
to derive the Gordon-Shapiro Dividend Model popularly used
to estimate the value of common stock. Make the reasonable
assumption that $i>g .$ Hint: What happens to $a^{n}$ as $n \rightarrow \infty$ if
$0<a<1 ?$ Source: Robert $D .$ Campbell.

Julian Wong

Numerade Educator

Problem 92

Alligator Teeth Researchers have developed a mathematical
model that can be used to estimate the number of teeth $N(t)$
at time $t$ (days of incubation) for Alligator mississippiensis,
where
$$N(t)=71.8 e^{-8.96 e^{-0.00035}}$$
$$\begin{array}{l}{\text { Source: Journal of Theoretical Biology. }} \\ {\text { (a) Find } N(65) \text { , the number of teeth of an alligator that hatched }} \\ {\text { after } 65 \text { days. }} \\ {\text { (b) Find } \lim _{t \rightarrow \infty} N(t) \text { and use this value as an estimate of the }} \\ {\text { number of teeth of a newborn alligator. (Hint: See Exer- }} \\ {\text { cise } 67 . \text { Does this estimate differ significantly from the }} \\ {\text { estimate of part (a)? }}\end{array}$$

Maria Dearborn

Numerade Educator

Problem 93

Sediment To develop strategies to manage water quality in
polluted lakes, biologists must determine the depths of sedi-
ments and the rate of sedimentation. It has been determined that
the depth of sediment $D(t)$ (in centimeters) with respect to time
(in years before 1990 ) for Lake Coeur d'Alene, Idaho, can be
estimated by the equation
$$D(t)=155\left(1-e^{-0.0133 t}\right)$$
$$\begin{array}{l}{\text { Source: The Mathematics Teacher. }} \\ {\text { (a) Find } D(20) \text { and interpret. }} \\ {\text { (b) Find } \lim _{t \rightarrow \infty} D(t) \text { and interpret. }}\end{array}$$

Julian Wong

Numerade Educator

Problem 94

Drug Concentration The concentration of a drug in a
patient's bloodstream $h$ hours after it was injected is given by
$$A(h)=\frac{0.17 h}{h^{2}+2.}$$
Find and interpret $\lim _{h \rightarrow \infty} A(h).$

Linh Vu

Numerade Educator

Problem 95

Legislative Voting Members of a legislature often must vote
repeatedly on the same bill. As time goes on, members may
change their votes. Suppose that $p_{0}$ is the probability that an
individual legislator favors an issue before the first roll call
vote and sunpose that $p$ is the probability of a change in posi-
tion from one vote to the next. Then the probability that the
legislator will vote "ves" on the $n$ th roll call is given by
$$p_{n}=\frac{1}{2}+\left(p_{0}-\frac{1}{2}\right)(1-2 p)^{n}$$
For example, the chance of a "yes" on the third roll call vote is
$$p_{3}=\frac{1}{2}+\left(p_{0}-\frac{1}{2}\right)(1-2 p)^{3.}$$
Source: Mathematics in the Behavioral and Social Sciences.
Suppose that there is a chance of $p_{0}=0.7$ that Congressman
Stephens will favor the budget appropriation bill before the first
roll call, but only a probability of $p=0.2$ that he will change his
mind on the subsequent vote. Find and interpret the following.
$$\begin{array}{ll}{\text { (a) } p_{2}} & {\text { (b) } p_{4}} \\ {\text { (c) } p_{8}} & {\text { (d) } \lim _{n \rightarrow \infty} p_{n}}\end{array}$$
YOUR TURN ANSWERS
$$\begin{array}{lll}{\text { 1. } 3} & {\text { 2. } 4} & {\text { 3.5 }}\end{array}$$
4. Does not exist.
$$\quad 5.-48/6 .-7 \quad 7.1 / 2 \quad \text { 8. } 1 / 3$$

Julian Wong

Numerade Educator