Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 9

The Derivative - all with Video Answers

Educators

Section 1

Limits

Problem 1

In Exercises 1-8, use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Amrita Bhasin

Amrita Bhasin

Numerade Educator

Problem 2

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 3

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 4

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 5

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 6

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 7

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 8

Use the graph of the given function $f$ to determine $\lim f(x)$ at the indicated value of $a$, if it exists.

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 9

In Exercises 9-16, complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=x^{2}+1 ; \lim _{x \rightarrow 2} f(x) \\
\hline x \quad 1.9 \quad 1.99 \quad 1.999 \quad 2.001 \quad 2.01 \quad 2.1 \\
\hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\
\hline
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 10

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=2 x^{2}-1 ; \lim _{x \rightarrow 1} f(x) \\
\hline x \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\
\hline f(\boldsymbol{x}) & & & & & & \\
\hline
\end{array}
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 11

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow 0} f(x)\\
\begin{array}{lllllll}
\hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\
\hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\
\hline
\end{array}
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 12

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{|x-1|}{x-1} ; \lim _{x \rightarrow 1} f(x) \\
\hline x \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\
\hline f(x)
\end{array}
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 13

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{1}{(x-1)^{2}} ; \lim _{x \rightarrow 1} f(x) \\
\hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\
\hline \boldsymbol{f}(\boldsymbol{x})
\end{array}
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 14

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{1}{x-2} ; \lim _{x \rightarrow 2} f(x) \\
\hline \boldsymbol{x} \quad 1.9 \quad 1.99 \quad 1.999 \quad 2.001 \quad 2.01 \quad 2.1 \\
\hline \boldsymbol{f}(\boldsymbol{x})
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 15

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{x^{2}+x-2}{x-1} ; \lim _{x \rightarrow 1} f(x) \\
\hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\
\hline \boldsymbol{f}(\boldsymbol{x}) \\
\hline
\end{array}
$$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 16

Complete the table by computing $f(x)$ at the given values of $x$. Use these results to estimate the indicated limit (if it exists).
$$
\begin{array}{l}
f(x)=\frac{x-1}{x-1} ; \lim _{x \rightarrow 1} f(x) \\
\hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\
\hline \boldsymbol{f}(\boldsymbol{x}) & \multicolumn{3}{|c} {} \\
\hline
\end{array}
$$

Dwijendra Rao

Numerade Educator

Problem 17

In Exercises 17-22, sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 0 \\ -1 & \text { if } x>0\end{array} \quad(a=0)\right.$

Aditya Sood

Numerade Educator

Problem 18

Sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array} \quad(a=3)\right.$

Dwijendra Rao

Numerade Educator

Problem 19

Sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}x & \text { if } x<1 \\ 0 & \text { if } x=1 \\ -x+2 & \text { if } x>1\end{array} \quad(a=1)\right.$

Carson Merrill

Numerade Educator

Problem 20

Sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}-2 x+4 & \text { if } x<1 \\ 4 & \text { if } x=1 \\ x^{2}+1 & \text { if } x>1\end{array} \quad(a=1)\right.$

Carson Merrill

Numerade Educator

Problem 21

Sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}|x| & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array} \quad(a=0)\right.$

Dwijendra Rao

Numerade Educator

Problem 22

Sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}|x-1| & \text { if } x \neq 1 \\ 0 & \text { if } x=1\end{array} \quad(a=1)\right.$

Dwijendra Rao

Numerade Educator

Problem 23

In Exercises 23-40, find the indicated limit.
$\lim _{x \rightarrow 2} 3$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 24

Find the indicated limit.
$\lim _{x \rightarrow-2}-3$

Dwijendra Rao

Numerade Educator

Problem 25

Find the indicated limit.
$\lim _{x \rightarrow 3} x$

Dwijendra Rao

Numerade Educator

Problem 26

Find the indicated limit.
$\lim _{x \rightarrow-2}-3 x$

Dwijendra Rao

Numerade Educator

Problem 27

Find the indicated limit.
$\lim _{x \rightarrow 1}\left(1-2 x^{2}\right)$

Dwijendra Rao

Numerade Educator

Problem 28

Find the indicated limit.
$\lim _{t \rightarrow 3}\left(4 t^{2}-2 t+1\right)$

Dwijendra Rao

Numerade Educator

Problem 29

Find the indicated limit.
$\lim _{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+x+2\right)$

Dwijendra Rao

Numerade Educator

Problem 30

Find the indicated limit.
$\lim _{x \rightarrow 0}\left(4 x^{5}-20 x^{2}+2 x+1\right)$

Dwijendra Rao

Numerade Educator

Problem 31

Find the indicated limit.
$\lim _{s \rightarrow 0}\left(2 s^{2}-1\right)(2 s+4)$

Dwijendra Rao

Numerade Educator

Problem 32

Find the indicated limit.
$\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(x^{2}-4\right)$

Dwijendra Rao

Numerade Educator

Problem 33

Find the indicated limit.
$\lim _{x \rightarrow 2} \frac{2 x+1}{x+2}$

Dwijendra Rao

Numerade Educator

Problem 34

Find the indicated limit.
$\lim _{x \rightarrow 1} \frac{x^{3}+1}{2 x^{3}+2}$

Dwijendra Rao

Numerade Educator

Problem 35

Find the indicated limit.
$\lim _{x \rightarrow 2} \sqrt{x+2}$

Dwijendra Rao

Numerade Educator

Problem 36

Find the indicated limit.
$\lim _{x \rightarrow-2} \sqrt[3]{5 x+2}$

Dwijendra Rao

Numerade Educator

Problem 37

Find the indicated limit.
$\lim _{x \rightarrow-3} \sqrt{2 x^{4}+x^{2}}$

Dwijendra Rao

Numerade Educator

Problem 38

Find the indicated limit.
$\lim _{x \rightarrow 2} \sqrt{\frac{2 x^{3}+4}{x^{2}+1}}$

Dwijendra Rao

Numerade Educator

Problem 39

Find the indicated limit.
$\lim _{x \rightarrow-1} \frac{\sqrt{x^{2}+8}}{2 x+4}$

Dwijendra Rao

Numerade Educator

Problem 40

Find the indicated limit.
$\lim _{x \rightarrow 3} \frac{x \sqrt{x^{2}+7}}{2 x-\sqrt{2 x+3}}$

Dwijendra Rao

Numerade Educator

Problem 41

In Exercises 41-48, find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a}[f(x)-g(x)]$

Linda Hand

Numerade Educator

Problem 42

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a} 2 f(x)$

Dwijendra Rao

Numerade Educator

Problem 43

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a}[2 f(x)-3 g(x)]$

Dwijendra Rao

Numerade Educator

Problem 44

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a}[f(x) g(x)]$

Dwijendra Rao

Numerade Educator

Problem 45

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a} \sqrt{g(x)}$

Dwijendra Rao

Numerade Educator

Problem 46

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a} \sqrt[3]{5 f(x)+3 g(x)}$

Dwijendra Rao

Numerade Educator

Problem 47

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)}$

Dwijendra Rao

Numerade Educator

Problem 48

Find the indicated limit given that $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=4$
$\lim _{x \rightarrow a} \frac{g(x)-f(x)}{f(x)+\sqrt{g(x)}}$

Dwijendra Rao

Numerade Educator

Problem 49

In Exercises 49-62, find the indicated limit, if it exists.
$\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 50

Find the indicated limit, if it exists.
$\lim _{x \rightarrow-2} \frac{x^{2}-4}{x+2}$

Dwijendra Rao

Numerade Educator

Problem 51

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 0} \frac{x^{2}-x}{x}$

Dwijendra Rao

Numerade Educator

Problem 52

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 0} \frac{2 x^{2}-3 x}{x}$

Dwijendra Rao

Numerade Educator

Problem 53

Find the indicated limit, if it exists.
$\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$

Dwijendra Rao

Numerade Educator

Problem 54

Find the indicated limit, if it exists.
$\lim _{b \rightarrow-3} \frac{b+1}{b+3}$

Dwijendra Rao

Numerade Educator

Problem 55

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 1} \frac{x}{x-1}$

Dwijendra Rao

Numerade Educator

Problem 56

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 2} \frac{x+2}{x-2}$

Dwijendra Rao

Numerade Educator

Problem 57

Find the indicated limit, if it exists.
$\lim _{x \rightarrow-2} \frac{x^{2}-x-6}{x^{2}+x-2}$

Melissa Munoz

Numerade Educator

Problem 58

Find the indicated limit, if it exists.
$\lim _{z \rightarrow 2} \frac{z^{3}-8}{z-2}$

Dwijendra Rao

Numerade Educator

Problem 59

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$
Hint: Multiply by $\frac{\sqrt{x}+1}{\sqrt{x}+1}$

Dwijendra Rao

Numerade Educator

Problem 60

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}$
Hint: See Exercise 59 .

Dwijendra Rao

Numerade Educator

Problem 61

Find the indicated limit, if it exists.
$\lim _{x \rightarrow 1} \frac{x-1}{x^{3}+x^{2}-2 x}$

Dwijendra Rao

Numerade Educator

Problem 62

Find the indicated limit, if it exists.
$\lim _{x \rightarrow-2} \frac{4-x^{2}}{2 x^{2}+x^{3}}$

Dwijendra Rao

Numerade Educator

Problem 63

In Exercises 63-68, use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Amy Jiang

Numerade Educator

Problem 64

Use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Dwijendra Rao

Numerade Educator

Problem 65

Use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Dwijendra Rao

Numerade Educator

Problem 66

Use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Dwijendra Rao

Numerade Educator

Problem 67

Use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Dwijendra Rao

Numerade Educator

Problem 68

Use the graph of the function $f$ to determine $\lim f(x)$ and $\lim f(x)$, if they exist.

Dwijendra Rao

Numerade Educator

Problem 69

In Exercises 69-72, complete the table by computing $f(x)$ at the given values of $x$. Use the results to guess at the indicated limits, if they exist.
$$
f(x)=\frac{1}{x^{2}+1} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x)
$$

Dwijendra Rao

Numerade Educator

Problem 70

Complete the table by computing $f(x)$ at the given values of $x$. Use the results to guess at the indicated limits, if they exist.
$$
f(x)=\frac{2 x}{x+1} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x)
$$

Dwijendra Rao

Numerade Educator

Problem 71

Complete the table by computing $f(x)$ at the given values of $x$. Use the results to guess at the indicated limits, if they exist.
$$
f(x)=3 x^{3}-x^{2}+10 ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x)
$$

Dwijendra Rao

Numerade Educator

Problem 72

Complete the table by computing $f(x)$ at the given values of $x$. Use the results to guess at the indicated limits, if they exist.
$$
f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x)
$$

Dwijendra Rao

Numerade Educator

Problem 73

In Exercises 73-80, find the indicated limits, if they exist.
$\lim _{x \rightarrow \infty} \frac{3 x+2}{x-5}$

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 74

Find the indicated limits, if they exist.
$\lim _{x \rightarrow-\infty} \frac{4 x^{2}-1}{x+2}$

Dwijendra Rao

Numerade Educator

Problem 75

Find the indicated limits, if they exist.
$\lim _{x \rightarrow-\infty} \frac{3 x^{3}+x^{2}+1}{x^{3}+1}$

Dwijendra Rao

Numerade Educator

Problem 76

Find the indicated limits, if they exist.
$\lim _{x \rightarrow \infty} \frac{2 x^{2}+3 x+1}{x^{4}-x^{2}}$

Dwijendra Rao

Numerade Educator

Problem 77

Find the indicated limits, if they exist.
$\lim _{x \rightarrow-\infty} \frac{x^{4}+1}{x^{3}-1}$

Dwijendra Rao

Numerade Educator

Problem 78

Find the indicated limits, if they exist.
$\lim _{x \rightarrow \infty} \frac{4 x^{4}-3 x^{2}+1}{2 x^{4}+x^{3}+x^{2}+x+1}$

Dwijendra Rao

Numerade Educator

Problem 79

Find the indicated limits, if they exist.
$\lim _{x \rightarrow \infty} \frac{x^{5}-x^{3}+x-1}{x^{6}+2 x^{2}+1}$

Dwijendra Rao

Numerade Educator

Problem 80

Find the indicated limits, if they exist.
$\lim _{x \rightarrow \infty} \frac{2 x^{2}-1}{x^{3}+x^{2}+1}$

Dwijendra Rao

Numerade Educator

Problem 81

ToxIc WASTE A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to city council members indicates that the cost, measured in millions of dollars, of removing $x \%$ of the toxic pollutant is given by
$$
C(x)=\frac{0.5 x}{100-x} \quad(0<x<100)
$$
a. Find the cost of removing $50 \%, 60 \%, 70 \%, 80 \%, 90 \%$, and $95 \%$ of the pollutant.
b. Evaluate
$$
\lim _{x \rightarrow 100} \frac{0.5 x}{100-x}
$$
and interpret your result.

Wendi Zhao

Numerade Educator

Problem 82

The population of a certain breed of rabbits introduced into an isolated island is given by
$$
P(t)=\frac{72}{9-t} \quad(0 \leq t<9)
$$
where $t$ is measured in months.
a. Find the number of rabbits present in the island initially (at $t=0$ ).
b. Show that the population of rabbits is increasing without bound.
c. Sketch the graph of the function $P$. (Comment: This phenomenon is referred to as a doomsday situation.)

Wendi Zhao

Numerade Educator

Problem 83

The average cost/disc in dollars incurred by Herald Records in pressing $x$ DVDs is given by the average cost function
$$
\bar{C}(x)=2.2+\frac{2500}{x}
$$
Evaluate $\lim _{x \rightarrow \infty} \bar{C}(x)$ and interpret your result.

Dwijendra Rao

Numerade Educator

Problem 84

The concentration of a certain drug in a patient's bloodstream $t \mathrm{hr}$ after injection is given by
$$
C(t)=\frac{0.2 t}{t^{2}+1}
$$
$\mathrm{mg} / \mathrm{cm}^{3}$. Evaluate $\lim _{t \rightarrow \infty} C(t)$ and interpret your result.

Dwijendra Rao

Numerade Educator

Problem 85

The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function
$$
T(x)=\frac{120 x^{2}}{x^{2}+4}
$$
where $T(x)$ is measured in millions of dollars and $x$ is the number of months since the movie's release.
a. What are the total box-office receipts after the first month? The second month? The third month?
b. What will the movie gross in the long run (when $x$ is very large)?

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 86

The percentage of a certain brand of computer chips that will fail after $t$ yr of use is estimated to be
$$
P(t)=100\left(1-e^{-0.1 t}\right)
$$
a. What percentage of this brand of computer chips are expected to be usable after 3 yr?
b. Evaluate $\lim _{t \rightarrow \infty} P(t)$. Did you expect this result?

Carson Merrill

Numerade Educator

Problem 87

A study of driving costs of 2008 mediumsized sedans found that the average cost (car payments, gas, insurance, upkeep, and depreciation), measured in cents/ mile, is approximated by the function
$$
C(x)=\frac{1735.2}{x^{1.72}}+38.6
$$
where $x$ denotes the number of miles (in thousands) the car is driven in a year.
a. What is the average cost of driving a medium-sized sedan $\begin{array}{lll}5000 \mathrm{mi} / \mathrm{yr} ? & 10,000 \mathrm{mi} / \mathrm{yr} ? & 15,000 \mathrm{mi} / \mathrm{yr} ? & 20,000 \mathrm{mi} / \mathrm{yr} ?\end{array}$
$25,000 \mathrm{mi} / \mathrm{yr} ?$
b. Use part (a) to sketch the graph of the function $C$.
c. What happens to the average cost as the number of miles driven increases without bound?

Wendi Zhao

Numerade Educator

Problem 88

The rate of production $R$ in photosynthesis is related to the light intensity $I$ by the function
$$
R(I)=\frac{a I}{b+I^{2}}
$$
where $a$ and $b$ are positive constants.
a. Taking $a=b=1$, compute $R(I)$ for $I=0,1,2,3,4$, and 5 .
b. Evaluate $\lim _{I \rightarrow \infty} R(I)$.
c. Use the results of parts (a) and (b) to sketch the graph of $R$. Interpret your results.

Wendi Zhao

Numerade Educator

Problem 89

In Exercises 89-94, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
If $\lim f(x)$ exists, then $f$ is defined at $x=a$.

Stephen Hobbs

Numerade Educator

Problem 90

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
If $\lim _{x \rightarrow 0} f(x)=4$ and $\lim _{x \rightarrow 0} g(x)=0$, then $\lim _{x \rightarrow 0} f(x) g(x)=0$.

William Semus

Numerade Educator

Problem 91

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
If $\lim _{x \rightarrow 2} f(x)=3$ and $\lim _{x \rightarrow 2} g(x)=0$, then $\lim _{x \rightarrow 2}[f(x)] /[g(x)]$
does not exist.

Wendi Zhao

Numerade Educator

Problem 92

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
If $\lim _{x \rightarrow 3} f(x)=0$ and $\lim _{x \rightarrow 3} g(x)=0$, then $\lim _{x \rightarrow 3}[f(x)] /[g(x)]$
does not exist.

Wendi Zhao

Numerade Educator

Problem 93

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
$\lim _{x \rightarrow 2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)=\lim _{x \rightarrow 2} \frac{x}{x+1}+\lim _{x \rightarrow 2} \frac{3}{x-1}$

Wendi Zhao

Numerade Educator

Problem 94

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
$\lim _{x \rightarrow 1}\left(\frac{2 x}{x-1}-\frac{2}{x-1}\right)=\lim _{x \rightarrow 1} \frac{2 x}{x-1}-\lim _{x \rightarrow 1} \frac{2}{x-1}$

Wendi Zhao

Numerade Educator

Problem 95

Certain proteins, known as enzymes, serve as catalysts for chemical reactions in living things. In 1913 Leonor Michaelis and L. M. Menten discovered the following formula giving the initial speed $V$ (in moles/liter/second) at which the reaction begins in terms of the amount of substrate $x$ (the substance heing acted upon, measured in moles/liters) present:
$$
V=\frac{a x}{x+b}
$$
where $a$ and $b$ are positive constants. Evaluate
$$
\lim _{x \rightarrow \infty} \frac{a x}{x+b}
$$
and interpret your result.

Wendi Zhao

Numerade Educator

Problem 96

Show by means of an example that $\lim _{x \rightarrow a}[f(x)+g(x)]$ may exist even though neither $\lim _{x \rightarrow a} f(x)$ nor $\lim _{x \rightarrow a} g(x)$ exists. Does this example contradict Theorem $1 ?$

Wendi Zhao

Numerade Educator

Problem 97

Show by means of an example that $\lim _{x \rightarrow a}[f(x) g(x)]$ may exist even though neither $\lim _{x \rightarrow a} f(x)$ nor $\lim _{x \rightarrow a} g(x)$ exists. Does this example contradict Theorem $1 ?$

Wendi Zhao

Numerade Educator

Problem 98

Show by means of an example that $\lim _{x \rightarrow a} f(x) / g(x)$ may exist even though neither $\lim _{x \rightarrow a} f(x)$ nor $\lim _{x \rightarrow a} g(x)$ exists. Does this example contradict Theorem $1 ?$

Wendi Zhao

Numerade Educator