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Finite Mathematics and Calculus with Applications

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

Chapter 11

The Derivative

Educators


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 $\lim _{x \rightarrow 2} f(x)$
a. is 5
b. is 6
c. does not exist.
d. is infinite.

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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 $\lim _{x \rightarrow 2} f(x)$
a. is $-1$
b. does not exist.
c. is infinite.
d. is 1.

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Problem 3

In Exercises 1–4, choose the best answer for each limit.
If $\lim _{x \rightarrow 4} f(x)=\lim _{x \rightarrow 4^{+}} f(x)=6,$ but $f(4)$ does not exist, then $\lim _{x \rightarrow 4} f(x)$
a. does not exist.
b. is 6
c. is $-\infty$
d. is $\infty$

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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 $\lim _{x \rightarrow 1} f(x)$
a. is $\infty$
b. is $-\infty$
c. does not exist.
d. is 1.

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

Use the table of values to estimate $\lim _{x \rightarrow 1} f(x)$
$$
\begin{array}{|c|cccccccc}{x} & {0.9} & {0.99} & {0.999} & {0.9999} & {1.0001} & {1.001} & {1.01} & {1.1} \\ \hline f(x) & {3.9} & {3.99} & {3.999} & {3.9999} & {4.0001} & {4.001} & {4.01} & {4.1}\end{array}
$$

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Problem 16

Complete the tables and use the results to find the indicated limits.
If $f(x)=2 x^{2}-4 x+7,$ find $\lim _{x \rightarrow 1} f(x)$
$$\begin{array}{|c|ccc}{x} & {0.9} & {0.99} & {0.999} & {1.001} & {1.01} & {1.1} \\ {f(x)} &&& {5.000002} & {5.000002}\end{array}$$

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Problem 17

Complete the tables and use the results to find the indicated limits.
If $k(x)=\frac{x^{3}-2 x-4}{x-2},$ find $\lim _{x \rightarrow 2} k(x)$
$$\begin{array}{|c|ccc}{x} & {1.9} & {1.99} & {1.999} & {2.001} & {2.01} & {2.1} \\ {k(x)} \end{array}$$

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Problem 18

Complete the tables and use the results to find the indicated limits.
If $f(x)=\frac{2 x^{3}+3 x^{2}-4 x-5}{x+1},$ find $\lim _{x \rightarrow-1} f(x)$
$$\begin{array}{|c|ccc}{x} & {-1.1} & {-1.01} & {-1.001} & {-0.999} & {-0.99 } & {-0.9} \\ {f(x)} \end{array}$$

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Problem 19

Complete the tables and use the results to find the indicated limits.
If $h(x)=\frac{\sqrt{x}-2}{x-1},$ find $\lim _{x \rightarrow 1} h(x)$
$$\begin{array}{|c|ccc}{x} & {0.9} & {0.99} & {0.999} & {1.001} & {1.01} & {1.1} \\ {h(x)} \end{array}$$

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Problem 20

Complete the tables and use the results to find the indicated limits.
If $f(x)=\frac{\sqrt{x}-3}{x-3},$ find $\lim _{x \rightarrow 3} f(x)$
$$\begin{array}{|c|ccc}{x} & {2.9} & {2.99} & {2.999} & {3.001} & {3.01} & {3.1} \\ {f(x)} \end{array}$$

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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.
$$
\lim _{x \rightarrow 4}[f(x)-g(x)]
$$

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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.
$$
\lim _{x \rightarrow 4}[g(x) \cdot f(x)]
$$

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

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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.
$$
\lim _{x \rightarrow 4} \log _{3} f(x)
$$

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

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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.
$$
\lim _{x \rightarrow 4} \sqrt[3]{g(x)}
$$

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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.
$$
\lim _{x \rightarrow 4} 2^{f(x)}
$$

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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.
$$
\lim _{x \rightarrow 4}[1+f(x)]^{2}
$$

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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.
$$
\lim _{x \rightarrow 4} \frac{f(x)+g(x)}{2 g(x)}
$$

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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.
$$
\lim _{x \rightarrow 4} \frac{5 g(x)+2}{1-f(x)}
$$

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Problem 31

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

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Problem 32

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

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Problem 33

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow 1} \frac{5 x^{2}-7 x+2}{x^{2}-1}
$$

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Problem 34

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+x-6}
$$

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Problem 35

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

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Problem 36

Use the properties of limits to help 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 37

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

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Problem 38

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

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Problem 39

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

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Problem 40

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow 36} \frac{\sqrt{x}-6}{x-36}
$$

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Problem 41

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

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Problem 42

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

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Problem 43

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow \infty} \frac{3 x}{7 x-1}
$$

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Problem 44

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

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Problem 45

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

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Problem 46

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

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Problem 47

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

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Problem 48

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

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Problem 49

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow \infty} \frac{2 x^{3}-x-3}{6 x^{2}-x-1}
$$

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Problem 50

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

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Problem 51

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
$$
\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x^{4}}{9 x^{2}+5 x-6}
$$

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Problem 52

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

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Problem 53

Let $f(x)=\left\{\begin{array}{ll}{x^{3}+2} & {\text { if } x \neq-1} \\ {5} & {\text { if } x=-1}\end{array} \quad \text { Find } \lim _{x \rightarrow-1} f(x)\right.$

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Problem 54

Let $g(x)=\left\{\begin{array}{ll}{0} & {\text { if } x=-2} \\ {\frac{1}{2} x^{2}-3} & {\text { if } x \neq-2}\end{array} \quad \text { Find } \lim _{x \rightarrow-2} g(x)\right.$

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Problem 55

Let $f(x)=\left\{\begin{array}{ll}{x-1} & {\text { if } x<3} \\ {2} & {\text { if } 3 \leq x \leq 5} \\ {x+3} & {\text { if } x > 5}\end{array}\right.$
a. Find $\lim _{x \rightarrow 3} f(x)$
b. Find $\lim _{x \rightarrow 5} f(x)$

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Problem 56

Let $g(x)=\left\{\begin{array}{ll}{5} & {\text { if } x<0} \\ {x^{2}-2} & {\text { if } 0 \leq x \leq 3} \\ {7} & {\text { if } x>3}\end{array}\right.$
a. Find $\lim _{x \rightarrow 0} g(x)$
b. Find $\lim _{x \rightarrow 3} g(x)$

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Problem 57

In Exercises 57–60, 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

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Problem 58

In Exercises 57–60, 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 32

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Problem 59

In Exercises 57–60, 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 33

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Problem 60

In Exercises 57–60, 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 34

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Problem 61

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

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Problem 62

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

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Problem 63

How can you tell that the graph in Figure 10 is more representative of the function $f(x)=1 /(x-1)$ than the graph in Figure 11$?$

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Problem 64

A friend who is confused about limits wonders why you investigate 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?

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Problem 65

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

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Problem 66

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

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Problem 67

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?

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Problem 68

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

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Problem 69

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

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Problem 70

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

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Problem 71

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 $x .$
$$
\lim _{x \rightarrow 1} \frac{x^{4}+4 x^{3}-9 x^{2}+7 x-3}{x-1}
$$

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Problem 72

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 $x .$
$$
\lim _{x \rightarrow 2} \frac{x^{4}+x-18}{x^{2}-4}
$$

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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 $x .$
$$
\lim _{x \rightarrow-1} \frac{x^{1 / 3}+1}{x+1}
$$

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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 $x .$
$$
\lim _{x \rightarrow 4} \frac{x^{3 / 2}-8}{x+x^{1 / 2}-6}
$$

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Problem 75

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 $f(x) .$
$$
\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{2}+5}}{2 x}
$$

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Problem 76

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 $f(x) .$
$$
\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{2}+5}}{2 x}
$$

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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 $f(x) .$
$$
\lim _{x \rightarrow-\infty} \frac{\sqrt{36 x^{2}+2 x+7}}{3 x}
$$

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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 $f(x) .$
$$
\lim _{x \rightarrow \infty} \frac{\sqrt{36 x^{2}+2 x+7}}{3 x}
$$

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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 $f(x) .$
$$
\lim _{x \rightarrow \infty} \frac{\left(1+5 x^{1 / 3}+2 x^{5 / 3}\right)^{3}}{x^{5}}
$$

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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 $f(x) .$
$$
\lim _{x \rightarrow-\infty} \frac{\left(1+5 x^{1 / 3}+2 x^{5 / 3}\right)^{3}}{x^{5}}
$$

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Problem 81

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

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Problem 82

When the price of an essential commodity (such as gasoline) rises rapidly, consumption drops slowly at first. If the price continues to rise, how- ever, a “tipping” point may be reached, at which consumption takes a sudden substantial drop. Suppose the accompanying graph shows the consumption of gasoline, in millions of gallons, 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. $\lim _{t \rightarrow 12} G(t)$
b. $\lim _{t \rightarrow 16} G(t)$
c. $G(16)$
d. The tipping point (in months)

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Problem 83

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 since it was first established in 1933. Let represent the sales tax per dollar spent in year x. Find
the following. Source: California State.
a. $\lim _{x \rightarrow 53} T(x)$
b. $\lim _{x \rightarrow 09^{-}} T(x)$
c. $\lim _{x \rightarrow 09^{+}} T(x)$
d. $\lim _{x \rightarrow 09} T(x)$
e. $T(09)$

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Problem 84

The graph below shows how the postage required to mail a letter in the United States has changed in recent years. Let be the cost to mail a letter in the year t. Find the following. Source: United States Postal Service.
a. $\lim _{t \rightarrow 2009^{-}} C(t)$
b. $\lim _{t \rightarrow 2009^{+}} C(t)$
c. $\lim _{t \rightarrow 2009} C(t)$
d. $C(2009)$

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Problem 85

The cost (in dollars) for manufacturing a particular DVD is
$$
C(x)=15,000+6 x
$$
where $x$ is the number of DVDs produced. Recall from the previous chapter that the average cost per DVD, denoted by $\overline{C}(x)$ is found by dividing $C(x)$ by $x$ . Find and interpret $\lim _{x \rightarrow \infty} \overline{C}(x)$ .

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Problem 86

Average cost In Chapter $1,$ we saw that the cost to fly $x$ miles on American Airlines could be approximated by the equation
$$
C(x)=0.0738 x+111.83
$$
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 $\lim _{x \rightarrow \infty} \overline{C}(x) .$ Source: American Airlines.

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Problem 87

A company training program has determined that, on the average, a new employee produces items per day after s days of on-the-job training, where
$$
P(s)=\frac{63 s}{s+8}
$$
Find and interpret $\lim _{s \rightarrow \infty} P(s)$

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Problem 88

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, called a perpetuity, receive payments that take the form of an annuity in that the amount of the payment never changes. However, normally the payments for preferred stock do not end but theoretically 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.

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Problem 89

For some annuities encountered in business finance, called growing annuities, the amount of the periodic 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. Compute 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.

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Problem 90

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
$$N(t)=71.8 e^{-896 e^{-0.0688}}$$
Source: Journal of Theoretical Biology.
a. Find $N(65)$ , the number of teeth of an alligator that hatched after 65 days.
b. Find $\lim _{t \rightarrow \infty} N(t)$ and use this value as an estimate of the number of teeth of a newborn alligator. (Hint: See Exercise $65 .$ ) Does this estimate differ significantly from the estimate
of part a?

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Problem 91

To develop strategies to manage water quality in polluted lakes, biologists must determine the depths of sediments 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
$$
D(t)=155\left(1-e^{-0.0133 t}\right)
$$
Source: Mathematics Teacher.
a. Find $D(20)$ and interpret.
b. Find $\lim _{t \rightarrow \infty} D(t)$ and interpret.

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Problem 92

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 $\lim _{h \rightarrow \infty} A(h)$

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Problem 93

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 suppose that $p$ is the probability of a change in position from one vote to the next. Then the probability that the legislator will vote "yes" 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.
$\begin{array}{ll}{\mathbf{a} \cdot p_{2}} & {\mathbf{b} \cdot p_{4}} \\ {\mathbf{c} \cdot p_{8}} & {\mathbf{d} \cdot \lim _{n \rightarrow \infty} p_{n}}\end{array}$

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