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Calculus for Business, Economics, Life Sciences, and Social Sciences 13th

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 2

Limits and the Derivative

Educators


Problem 1

In Problems $1-8$, factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{2}-81
$$

Dwijendra R.
Numerade Educator

Problem 2

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{2}-64
$$

Dwijendra R.
Numerade Educator

Problem 3

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{2}-4 x-21
$$

Dwijendra R.
Numerade Educator

Problem 4

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{2}+5 x-36
$$

Dwijendra R.
Numerade Educator

Problem 5

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{3}-7 x^{2}+12 x
$$

Dwijendra R.
Numerade Educator

Problem 6

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
x^{3}+15 x^{2}+50 x
$$

Dwijendra R.
Numerade Educator

Problem 7

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
6 x^{2}-x-1
$$

Dwijendra R.
Numerade Educator

Problem 8

Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3).
$$
20 x^{2}+11 x-3
$$

Dwijendra R.
Numerade Educator

Problem 9

In Problems $9-16,$ use the graph of the function $f$ shown to estimate the indicated limits and function values.
$$
f(-0.5)
$$

Dwijendra R.
Numerade Educator

Problem 10

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
$$
f(-1.5)
$$

Dwijendra R.
Numerade Educator

Problem 11

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
$$
f(1.75)
$$

Dwijendra R.
Numerade Educator

Problem 12

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
$$
f(1.25)
$$

Dwijendra R.
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Problem 13

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 0^{-}} f(x)$
(B) $\lim _{x \rightarrow 0^{+}} f(x)$
(C) $\lim _{x \rightarrow 0} f(x)$
(D) $f(0)$

Dwijendra R.
Numerade Educator

Problem 14

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 1^{-}} f(x)$
(B) $\lim _{x \rightarrow 1^{+}} f(x)$
(C) $\lim _{i} f(x)$
(D) $f(1)$

Dwijendra R.
Numerade Educator

Problem 15

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 2^{-}} f(x)$
(B) $\lim _{x \rightarrow 2^{+}} f(x)$
(C) $\lim _{x \rightarrow 2} f(x)$
(D) $f(2)$
(E) Is it possible to redefine $f(2)$ so that $\lim _{x \rightarrow 2} f(x)=f(2)$ ? Explain.

Dwijendra R.
Numerade Educator

Problem 16

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 4^{-}} f(x)$
(B) $\lim _{x \rightarrow 4^{+}} f(x)$
(C) $\lim _{x \rightarrow 4} f(x)$
(D) $f(4)$
(E) Is it possible to define $f(4)$ so that $\lim _{x \rightarrow 4} f(x)=f(4)$ ? Explain.

Dwijendra R.
Numerade Educator

Problem 17

In Problems $17-24$, use the graph of the function $g$ shown to estimate the indicated limits and function values.
$$
g(1.9)
$$

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Problem 18

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
$$
g(0.1)
$$

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Problem 19

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
$$
g(3.5)
$$

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Problem 20

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
$$
g(2.5)
$$

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Problem 21

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 1^{-}} g(x)$
(B) $\lim _{x \rightarrow 1^{+}} g(x)$
(C) $\lim _{x \rightarrow 1} g(x)$
(D) $g(1)$
(E) Is it possible to define $g(1)$ so that $\lim _{x \rightarrow 1} g(x)=g(1) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 22

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 2^{-}} g(x)$
(B) $\lim _{x \rightarrow 2^{+}} g(x)$
(C) $\lim _{x \rightarrow 2} g(x)$
(D) $g(2)$

Dwijendra R.
Numerade Educator

Problem 23

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 3^{-}} g(x)$
(B) $\lim _{x \rightarrow 3^{+}} g(x)$
(C) $\lim _{x \rightarrow 3} g(x)$
(D) $g(3)$
(E) Is it possible to redefine $g(3)$ so that $\lim _{x \rightarrow 3} g(x)=g(3) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 24

Use the graph of the function $g$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 4^{-}} g(x)$
(B) $\lim _{x \rightarrow 4^{+}} g(x)$
(C) $\lim _{x \rightarrow 4} g(x)$
(D) $g(4)$

Dwijendra R.
Numerade Educator

Problem 25

In Problems $25-28,$ use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow-3^{+}} f(x)$
(B) $\lim _{x \rightarrow-3^{-}} f(x)$
(C) $\lim _{x \rightarrow-3} f(x)$
(D) $f(-3)$
(E) Is it possible to redefine $f(-3)$ so that $\lim _{x \rightarrow-3} f(x)=f(-3) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 26

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow-2^{+}} f(x)$
(B) $\lim _{x \rightarrow-2^{-}} f(x)$
(C) $\lim _{x \rightarrow-2} f(x)$
(D) $f(-2)$
(E) Is it possible to redefine $f(-2)$ so that $\lim _{x \rightarrow-2} f(x)=f(-2) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 27

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 0^{+}} f(x)$
(B) $\lim _{x \rightarrow 0^{-}} f(x)$
(C) $\lim _{x \rightarrow 0} f(x)$
(D) $f(0)$
(E) Is it possible to define $f(0)$ so that $\lim _{x \rightarrow 0} f(x)=f(0) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 28

Use the graph of the function $f$ shown to estimate the indicated limits and function values.
(A) $\lim _{x \rightarrow 2^{+}} f(x)$
(B) $\lim _{x \rightarrow 2^{-}} f(x)$
(C) $\lim _{x \rightarrow 2} f(x)$
(D) $f(2)$
(E) Is it possible to redefine $f(2)$ so that $\lim _{x \rightarrow 2} f(x)=f(2) ?$ Explain.

Dwijendra R.
Numerade Educator

Problem 29

In Problems 29-38, find each limit if it exists.
$$
\lim _{x \rightarrow 3} 4 x
$$

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Numerade Educator

Problem 30

Find each limit if it exists.
$$
\lim _{x \rightarrow-2} 3 x
$$

Dwijendra R.
Numerade Educator

Problem 31

Find each limit if it exists.
$$
\lim _{x \rightarrow-4}(x+5)
$$

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Numerade Educator

Problem 32

Find each limit if it exists.
$$
\lim _{x \rightarrow 5}(x-3)
$$

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Numerade Educator

Problem 33

Find each limit if it exists.
$$
\lim _{x \rightarrow 2} x(x-4)
$$

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Problem 34

Find each limit if it exists.
$$
\lim _{x \rightarrow-1} x(x+3)
$$

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Problem 35

Find each limit if it exists.
$$
\lim _{x \rightarrow-3} \frac{x}{x+5}
$$

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Problem 36

Find each limit if it exists.
$$
\lim _{x \rightarrow 4} \frac{x-2}{x}
$$

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Problem 37

Find each limit if it exists.
$$
\lim _{x \rightarrow 1} \sqrt{5 x+4}
$$

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Problem 38

Find each limit if it exists.
$$
\lim _{x \rightarrow 0} \sqrt{16-7 x}
$$

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Problem 39

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits in Problems $39-46 .$
$$
\lim _{x \rightarrow 1}(-3) f(x)
$$

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Problem 40

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1} 2 g(x)
$$

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Problem 41

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1}[2 f(x)+g(x)]
$$

Dwijendra R.
Numerade Educator

Problem 42

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1}[g(x)-3 f(x)]
$$

Dwijendra R.
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Problem 43

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1} \frac{2-f(x)}{x+g(x)}
$$

Dwijendra R.
Numerade Educator

Problem 44

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1} \frac{3-f(x)}{1-4 g(x)}
$$

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Problem 45

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1} \sqrt{g(x)-f(x)}
$$

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Numerade Educator

Problem 46

Given that $\lim _{x \rightarrow 1} f(x)=-5$ and $\lim _{x \rightarrow 1} g(x)=4,$ find the indicated limits.
$$
\lim _{x \rightarrow 1} \sqrt[3]{2 x+2 f(x)}
$$

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Numerade Educator

Problem 47

In Problems $47-50$, sketch a possible graph of a function that satisfies the given conditions.
$$
f(0)=1 ; \lim _{x \rightarrow 0^{-}} f(x)=3 ; \lim _{x \rightarrow 0^{+}} f(x)=1
$$

Dwijendra R.
Numerade Educator

Problem 48

Sketch a possible graph of a function that satisfies the given conditions.
$$
f(1)=-2 ; \lim _{x \rightarrow 1^{-}} f(x)=2 ; \lim _{x \rightarrow 1^{+}} f(x)=-2
$$

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Problem 49

Sketch a possible graph of a function that satisfies the given conditions.
$$
f(-2)=2 ; \lim _{x \rightarrow-2^{-}} f(x)=1 ; \lim _{x \rightarrow-2^{+}} f(x)=1
$$

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Numerade Educator

Problem 50

Sketch a possible graph of a function that satisfies the given conditions.
$$
f(0)=-1 ; \lim _{x \rightarrow 0^{-}} f(x)=2 ; \lim _{x \rightarrow 0^{+}} f(x)=2
$$

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Problem 51

In Problems $51-66$, find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{ll}1-x^{2} & \text { if } x \leq 0 \\ 1+x^{2} & \text { if } x>0\end{array}\right.$. Find
(A) $\lim _{x \rightarrow 0^{+}} f(x)$
(B) $\lim _{x \rightarrow 0^{-}} f(x)$
(C) $\lim _{x \rightarrow 0} f(x)$
(D) $f(0)$

Dwijendra R.
Numerade Educator

Problem 52

Find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{ll}2+x & \text { if } x \leq 0 \\ 2-x & \text { if } x>0\end{array}\right.$. Find
(A) $\lim _{x \rightarrow 0^{+}} f(x)$
(B) $\lim _{x \rightarrow 0^{-}} f(x)$
(C) $\lim _{x \rightarrow 0} f(x)$
(D) $f(0)$

Dwijendra R.
Numerade Educator

Problem 53

Find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<1 \\ 2 x & \text { if } x>1\end{array} .\right.$ Find
(A) $\lim _{x \rightarrow 1^{+}} f(x)$
(B) $\lim _{x \rightarrow 1^{-}} f(x)$
(C) $\lim _{x \rightarrow 1} f(x)$
(D) $f(1)$

Dwijendra R.
Numerade Educator

Problem 54

Find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{cl}x+3 & \text { if } x<-2 \\ \sqrt{x+2} & \text { if } x>-2\end{array}\right.$. Find
(A) $\lim _{x \rightarrow-2^{+}} f(x)$
(B) $\lim _{x \rightarrow-2^{-}} f(x)$
(C) $\lim _{x \rightarrow-2} f(x)$
(D) $f(-2)$

Dwijendra R.
Numerade Educator

Problem 55

Find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x+3} & \text { if } x<0 \\ \frac{x^{2}-9}{x-3} & \text { if } x>0\end{array}\right.$. Find
(A) $\lim _{x \rightarrow-3} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$

Dwijendra R.
Numerade Educator

Problem 56

Find each indicated quantity if it exists.
Let $f(x)=\left\{\begin{array}{ll}\frac{x}{x+3} & \text { if } x<0 \\ \frac{x}{x-3} & \text { if } x>0\end{array}\right.$. Find
(A) $\lim _{x \rightarrow-3} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$

Dwijendra R.
Numerade Educator

Problem 57

Find each indicated quantity if it exists.
Let $f(x)=\frac{|x-1|}{x-1} .$ Find
(A) $\lim _{x \rightarrow 1^{+}} f(x)$
(B) $\lim _{x \rightarrow 1^{-}} f(x)$
(C) $\lim _{x \rightarrow 1} f(x)$
(D) $f(1)$

Dwijendra R.
Numerade Educator

Problem 58

Find each indicated quantity if it exists.
Let $f(x)=\frac{x-3}{|x-3|} .$ Find
(A) $\lim _{x \rightarrow 3^{+}} f(x)$
(B) $\lim _{x \rightarrow 3^{-}} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$
(D) $f(3)$

Dwijendra R.
Numerade Educator

Problem 59

Let $f(x)=\frac{x-3}{|x-3|} .$ Find
(A) $\lim _{x \rightarrow 3^{+}} f(x)$
(B) $\lim _{x \rightarrow 3^{-}} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$
(D) $f(3)$

Dwijendra R.
Numerade Educator

Problem 60

Find each indicated quantity if it exists.
Let $f(x)=\frac{x+3}{x^{2}+3 x} .$ Find
(A) $\lim _{x \rightarrow-3} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$

Dwijendra R.
Numerade Educator

Problem 61

Find each indicated quantity if it exists.
Let $f(x)=\frac{x^{2}-x-6}{x+2} .$ Find
(A) $\lim _{x \rightarrow-2} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 3} f(x)$

Dwijendra R.
Numerade Educator

Problem 62

Find each indicated quantity if it exists.
Let $f(x)=\frac{x^{2}+x-6}{x+3}$. Find
(A) $\lim _{x \rightarrow-3} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 2} f(x)$

Dwijendra R.
Numerade Educator

Problem 63

Find each indicated quantity if it exists.
Let $f(x)=\frac{(x+2)^{2}}{x^{2}-4}$. Find
(A) $\lim _{x \rightarrow-2} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 2} f(x)$

Dwijendra R.
Numerade Educator

Problem 64

Find each indicated quantity if it exists.
Let $f(x)=\frac{x^{2}-1}{(x+1)^{2}} .$ Find
(A) $\lim _{x \rightarrow-1} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 1} f(x)$

Dwijendra R.
Numerade Educator

Problem 65

Find each indicated quantity if it exists.
Let $f(x)=\frac{2 x^{2}-3 x-2}{x^{2}+x-6} .$ Find
(A) $\lim _{x \rightarrow 2} f(x)$
(B) $\lim _{x \rightarrow 0} f(x)$
(C) $\lim _{x \rightarrow 1} f(x)$

Dwijendra R.
Numerade Educator

Problem 66

Find each indicated quantity if it exists.
Let $f(x)=\frac{3 x^{2}+2 x-1}{x^{2}+3 x+2}$. Find
(A) $\lim _{x \rightarrow-3} f(x)$
(B) $\lim _{x \rightarrow-1} f(x)$
(C) $\lim _{x \rightarrow 2} f(x)$

Dwijendra R.
Numerade Educator

Problem 67

In Problems 67-72, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $\lim _{x \rightarrow 1} f(x)=0$ and $\lim _{x \rightarrow 1} g(x)=0,$ then $\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}=0$.

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Problem 68

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $\lim _{x \rightarrow 1} f(x)=1$ and $\lim _{x \rightarrow 1} g(x)=1,$ then $\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}=1$.

Dwijendra R.
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Problem 69

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $f$ is a polynomial, then, as $x$ approaches 0 , the right-hand limit exists and is equal to the left-hand limit.

Dwijendra R.
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Problem 70

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $f$ is a rational function, then, as $x$ approaches 0 , the right hand limit exists and is equal to the left-hand limit.

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Problem 71

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $f$ is a function such that $\lim _{x \rightarrow 0} f(x)$ exists, then $f(0)$ exists.

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Problem 72

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
If $f$ is a function such that $f(0)$ exists, then $\lim _{x \rightarrow 0} f(x)$ exists.

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Problem 73

In Problems $73-80$, is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 7} \frac{(x-7)^{2}}{x^{2}-4 x-21}
$$

Dwijendra R.
Numerade Educator

Problem 74

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 2} \frac{x-5}{x+2}
$$

Dwijendra R.
Numerade Educator

Problem 75

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 4} \frac{x^{2}+4}{(x+4)^{2}}
$$

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Problem 76

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 9} \frac{x^{2}-5 x-36}{x-9}
$$

Dwijendra R.
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Problem 77

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow-6} \frac{x^{2}+36}{x+6}
$$

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Problem 78

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 10} \frac{x^{2}-15 x+50}{(x-10)^{2}}
$$

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Problem 79

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow 8} \frac{x-8}{x^{2}-64}
$$

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Problem 80

Is the limit expression a $0 / 0$ indeterminate form? Find the limit or explain why the limit does not exist.
$$
\lim _{x \rightarrow-3} \frac{x+3}{x-3}
$$

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Problem 81

Compute the following limit for each function in Problems $81-88 .$
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=3 x+1
$$

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Problem 82

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=5 x-1
$$

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Problem 83

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=x^{2}+1
$$

Dwijendra R.
Numerade Educator

Problem 84

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=x^{2}-2
$$

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Numerade Educator

Problem 85

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=-7 x+9
$$

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Problem 86

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=-4 x+13
$$

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Numerade Educator

Problem 87

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=|x+1|
$$

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Problem 88

Compute the following limit for each function.
$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$
$$
f(x)=-3|x|
$$

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Problem 89

Let $f$ be defined by
$$f(x)=\left\{\begin{array}{ll}1+m x & \text { if } x \leq 1 \\4-m x & \text { if } x>1\end{array}\right.$$
where $m$ is a constant.
(A) Graph $f$ for $m=1,$ and find
$$\lim _{x \rightarrow 1^{-}} f(x) \quad \text { and } \quad \lim _{x \rightarrow 1^{+}} f(x)$$
(B) Graph $f$ for $m=2,$ and find
$$\lim _{x \rightarrow 1^{-}} f(x) \quad \text { and } \quad \lim _{x \rightarrow 1^{+}} f(x)$$
(C) Find $m$ so that
$$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$$
and graph $f$ for this value of $m$.
(D) Write a brief verbal description of each graph. How does the graph in part (C) differ from the graphs in parts (A) and $(\mathrm{B}) ?$

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Problem 90

Let $f$ be defined by
$$f(x)=\left\{\begin{array}{cl}-3 m+0.5 x & \text { if } x \leq 2 \\3 m-x & \text { if } x>2\end{array}\right.$$
where $m$ is a constant.
(A) Graph $f$ for $m=0$, and find
$$\lim _{x \rightarrow 2^{-}} f(x) \text { and } \lim _{x \rightarrow 2^{+}} f(x)$$
(B) Graph $f$ for $m=1$, and find
$$\lim _{x \rightarrow 2^{-}} f(x) \text { and } \lim _{x \rightarrow+} f(x)$$
(C) Find $m$ so that
$$\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)$$
and graph $f$ for this value of $m$.
(D) Write a brief verbal description of each graph. How does the graph in part (C) differ from the graphs in parts (A) and (B)?

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Problem 91

A long-distance telephone service charges $\$ 0.99$ for the first 20 minutes or less of a call and $\$ 0.07$ per minute thereafter.
(A) Write a piecewise definition of the charge $F(x)$ for a long-distance call lasting $x$ minutes.
(B) Graph $F(x)$ for $0<x \leq 40$.
(C) Find $\lim _{x \rightarrow 20^{-}} F(x), \lim _{x \rightarrow 20^{+}} F(x),$ and $\lim _{x \rightarrow 20} F(x),$ which-
ever exist.

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Problem 92

A second long-distance telephone service charges $\$ 0.09$ per minute for calls lasting 10 minutes or more and $\$ 0.18$ per minute for calls lasting less than 10 minutes.
(A) Write a piecewise definition of the charge $G(x)$ for a long-distance call lasting $x$ minutes.
(B) Graph $G(x)$ for $0<x \leq 40$
(C) Find $\lim _{x \rightarrow 10^{-}} G(x), \lim _{x \rightarrow 10^{+}} G(x),$ and $\lim _{x \rightarrow 10} G(x),$ which-ever exist.

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Problem 93

Refer to Problems 91 and $92 .$ Write a brief verbal comparison of the two services described for calls lasting 20 minutes or less.

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Problem 94

Refer to Problems 91 and $92 .$ Write a brief verbal comparison of the two services described for calls lasting more than 20 minutes.

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Problem 95

A company sells custom embroidered apparel and promotional products. Table 1 shows the volume discounts offered by the company, where $x$ is the volume of a purchase in dollars. Problems 95 and 96 deal with two different interpretations of this discount method.
Assume that the volume discounts in Table 1 apply to the entire purchase. That is, if the volume $x$ satisfies $\$ 300 \leq x<\$ 1,000,$ then the entire purchase is discounted $3 \%$. If the volume $x$ satisfies $\$ 1,000 \leq x<\$ 3,000$ the entire purchase is discounted $5 \%,$ and so on.
(A) If $x$ is the volume of a purchase before the discount is applied, then write a piecewise definition for the discounted price $D(x)$ of this purchase.
(B) Use one-sided limits to investigate the limit of $D(x)$ as $x$ approaches $\$ 1,000 .$ As $x$ approaches $\$ 3,000$

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Problem 96

A company sells custom embroidered apparel and promotional products. Table 1 shows the volume discounts offered by the company, where $x$ is the volume of a purchase in dollars. deal with two different interpretations of this discount method.
Assume that the volume discounts in Table 1 apply only to that portion of the volume in each interval. That is, the discounted price for a $\$ 4,000$ purchase would be computed as follows:
$$300+0.97(700)+0.95(2,000)+0.93(1,000)=3,809$$
(A) If $x$ is the volume of a purchase before the discount is applied, then write a piecewise definition for the discounted price $P(x)$ of this purchase.
(B) Use one-sided limits to investigate the limit of $P(x)$ as $x$ approaches $\$ 1,000 .$ As $x$ approaches $\$ 3,000$
(C) Compare this discount method with the one in Problem
95. Does one always produce a lower price than the other? Discuss.

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Problem 97

A state charges polluters an annual fee of $\$ 20$ per ton for each ton of pollutant emitted into the atmosphere, up to a maximum of 4,000 tons. No fees are charged for emissions beyond the 4,000 -ton limit. Write a piecewise definition of the fees $F(x)$ charged for the emission of $x$ tons of pollutant in a year. What is the limit of $F(x)$ as $x$ approaches 4,000 tons? As $x$ approaches 8,000 tons?

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Problem 98

Refer to Problem 97 . The average fee per ton of pollution is given by $A(x)=F(x) / x$. Write a piecewise definition of $A(x)$. What is the limit of $A(x)$ as $x$ approaches 4,000 tons? As $x$ approaches 8,000 tons?

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Problem 99

Statisticians often use piecewise-defined functions to predict outcomes of elections. For the following functions $f$ and $g,$ find the limit of each function as $x$ approaches 5 and as $x$ approaches 10
$$
\begin{array}{l}
f(x)=\left\{\begin{array}{cl}
0 & \text { if } x \leq 5 \\
0.8-0.08 x & \text { if } 5 < x < 10 \\
0 & \text { if } 10 \leq x
\end{array}\right. \\
g(x)=\left\{\begin{array}{cl}
0 & \text { if } x \leq 5 \\
0.8 x-0.04 x^{2}-3 & \text { if } 5 < x < 10 \\
1 & \text { if } 10 \leq x
\end{array}\right.
\end{array}
$$

Dwijendra R.
Numerade Educator