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Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 2

Limits And Continuity

Educators


Problem 1

For the function $g(x)$ graphed here, find the following limits or explain why they do not exist.
$\begin{array}{llll}{\text { a. } \lim _{x \rightarrow 1} g(x)} & {\text { b. } \lim _{x \rightarrow 2} g(x)} & {\text { c. } \lim _{x \rightarrow 3} g(x)}\end{array}$

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Problem 2

For the function $f(t)$ graphed here, find the following limits or ex- plain why they do not exist.
$\begin{array}{llll}{\text { a. } \lim _{t \rightarrow-2} f(t)} & {\text { b. } \lim _{t \rightarrow-1} f(t)} & {\text { c. } \lim _{t \rightarrow 0} f(t)}\end{array}$

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Problem 3

Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
a. $\lim _{x \rightarrow 0} f(x)$ exists.
b. $\lim _{x \rightarrow 0} f(x)=0$
c. $\lim _{x \rightarrow 0} f(x)=1$
d. $\lim _{x \rightarrow 1} f(x)=1$
e. $\lim _{x \rightarrow 1} f(x)=0$
f. $\lim _{x \rightarrow x_{0}} f(x)$ exists at every point $x_{0}$ in $(-1,1)$

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Problem 4

Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
a. $\lim _{x \rightarrow 2} f(x)$ does not exist.
b. $\lim _{x \rightarrow 2} f(x)=2$
c. $\lim _{x \rightarrow 1} f(x)$ does not exist.
d. $\lim _{x \rightarrow x_{0}} f(x)$ exists at every point $x_{0}$ in $(-1,1)$ .
e. $\lim _{x \rightarrow x_{0}} f(x)$ exists at every point $x_{0}$ in $(1,3)$

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Problem 5

In Exercises 5 and 6, explain why the limits do not exist.
$$
\lim _{x \rightarrow 0} \frac{x}{|x|}
$$

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Problem 6

In Exercises 5 and 6, explain why the limits do not exist.
$$
\lim _{x \rightarrow 1} \frac{1}{x-1}
$$

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Problem 7

Suppose that a function $f(x)$ is defined for all real values of $x$ except $x=x_{0} .$ Can anything be said about the existence of $\lim _{x \rightarrow x_{0}} f(x) ?$ Give reasons for your answer.

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Problem 8

Suppose that a function $f(x)$ is defined for all $x$ in $[-1,1] .$ Can anything be said about the existence of $\lim _{x \rightarrow 0} f(x) ?$ Give reasons for your answer.

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Problem 9

If $\lim _{x \rightarrow 1} f(x)=5,$ must $f$ be defined at $x=1 ?$ If it is, must $f(1)=5 ?$ Can we conclude anything about the values of $f$ at $x=1 ?$ Explain.

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Problem 10

If $f(1)=5,$ must $\lim _{x \rightarrow 1} f(x)$ exist? If it does, then must $\lim _{x \rightarrow 1} f(x)=5 ?$ Can we conclude anything about $\lim _{x \rightarrow 1} f(x) ?$ Explain.

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Problem 11

You will find a graphing calculator useful for Exercises 11–20.
Let $f(x)=\left(x^{2}-9\right) /(x+3)$
a. Make a table of the values of $f$ at the points $x=-3.1$ $-3.01,-3.001,$ and so on as far as your calculator can go. Then estimate $\lim _{x \rightarrow-3} f(x) .$ What estimate do you arrive at if you evaluate $f$ at $x=-2.9,-2.99,-2.999, \ldots$ instead?
b. Support your conclusions in part (a) by graphing $f$ near $x_{0}=-3$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-3$ .
c. Find $\lim _{x \rightarrow-3} f(x)$ algebraically, as in Example 5 .

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Problem 12

You will find a graphing calculator useful for Exercises 11–20.
Let $g(x)=\left(x^{2}-2\right) /(x-\sqrt{2})$
a. Make a table of the values of $g$ at the points $x=1.4,1.41$ $1.414,$ and so on through successive decimal approximations of $\sqrt{2} .$ Estimate $\lim _{x \rightarrow \sqrt{2}} g(x)$
b. Support your conclusion in part (a) by graphing $g$ near $x_{0}=\sqrt{2}$ and using $Z$ oom and Trace to estimate $y$ -values on the graph as $x \rightarrow \sqrt{2}$ .
c. Find $\lim _{x \rightarrow \sqrt{2}} g(x)$ algebraically.

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Problem 13

You will find a graphing calculator useful for Exercises 11–20.
Let $G(x)=(x+6) /\left(x^{2}+4 x-12\right)$
a. Make a table of the values of $G$ at $x=-5.9,-5.99,-5.999,$ and so on. Then estimate $\lim _{x \rightarrow-6} G(x) .$ What estimate do you arrive at if you evaluate $G$ at $x=-6.1,-6.01,-6.001, \ldots$ instead?
b. Support your conclusions in part (a) by graphing $G$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-6$ .
c. Find $\lim _{x \rightarrow-6} G(x)$ algebraically.

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Problem 14

You will find a graphing calculator useful for Exercises 11–20.
Let $h(x)=\left(x^{2}-2 x-3\right) /\left(x^{2}-4 x+3\right)$
a. Make a table of the values of $h$ at $x=2.9,2.99,2.999,$ and so on. Then estimate $\lim _{x \rightarrow 3} h(x)$ . What estimate do you arrive at if you evaluate $h$ at $x=3.1,3.01,3.001, \ldots$ instead?
b. Support your conclusions in part (a) by graphing $h$ near $x_{0}=3$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow 3$ .
c. Find $\lim _{x \rightarrow 3} h(x)$ algebraically.

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Problem 15

You will find a graphing calculator useful for Exercises 11–20.
Let $f(x)=\left(x^{2}-1\right) /(|x|-1)$
a. Make tables of the values of $f$ at values of $x$ that approach $x_{0}=-1$ from above and below. Then estimate $\lim _{x \rightarrow-1} f(x)$
b. Support your conclusion in part (a) by graphing $f$ near $\quad x_{0}=-1$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-1 .$
c. Find $\lim _{x \rightarrow-1} f(x)$ algebraically.

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Problem 16

You will find a graphing calculator useful for Exercises 11–20.
Let $F(x)=\left(x^{2}+3 x+2\right) /(2-|x|)$
a. Make tables of values of $F$ at values of $x$ that approach $x_{0}=-2$ from above and below. Then estimate $\lim _{x \rightarrow-2} F(x)$
b. Support your conclusion in part (a) by graphing $F$ near $x_{0}=-2$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-2$
c. Find $\lim _{x \rightarrow-2} F(x)$ algebraically..

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Problem 17

You will find a graphing calculator useful for Exercises 11–20.
Let $g(\theta)=(\sin \theta) / \theta$
a. Make a table of the values of $g$ at values of $\theta$ that approach $\theta_{0}=0$ from above and below. Then estimate $\lim _{\theta \rightarrow 0} g(\theta) .$
b. Support your conclusion in part (a) by graphing $g$ near $\theta_{0}=0$

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Problem 18

You will find a graphing calculator useful for Exercises 11–20.
Let $G(t)=(1-\cos t) / t^{2}$
a. Make tables of values of $G$ at values of $t$ that approach $t_{0}=0$ from above and below. Then estimate $\lim _{t \rightarrow 0} G(t) .$
b. Support your conclusion in part (a) by graphing $G$ near $t_{0}=0$

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Problem 19

You will find a graphing calculator useful for Exercises 11–20.
Let $f(x)=x^{1 /(1-x)}$
a. Make tables of values of $f$ at values of $x$ that approach $x_{0}=1$ $\quad$ from above and below. Does $f$ appear to have a limit as $x \rightarrow 1 ?$ If so, what is it? If not, why not?
b. Support your conclusions in part (a) by graphing $f$ near $x_{0}=1$

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Problem 20

You will find a graphing calculator useful for Exercises 11–20.
Let $f(x)=\left(3^{x}-1\right) / x$
a. Make tables of values of $f$ at values of $x$ that approach $x_{0}=0$ $\quad$ from above and below. Does $f$ appear to have a limit as $x \rightarrow 0 ?$ If so, what is it? If not, why not?
b. Support your conclusions in part (a) by graphing $f$ near $x_{0}=0$

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Problem 21

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow 2} 2 x
$$

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Problem 22

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow 0} 2 x
$$

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Problem 23

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow 1 / 3}(3 x-1)
$$

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Problem 24

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow 1} \frac{-1}{(3 x-1)}
$$

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Problem 25

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow-1} 3 x(2 x-1)
$$

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Problem 26

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow-1} \frac{3 x^{2}}{2 x-1}
$$

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Problem 27

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow \pi / 2} x \sin x
$$

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Problem 28

In Exercises 21–28, find the limits by substitution.
$$
\lim _{x \rightarrow \pi} \frac{\cos x}{1-\pi}
$$

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Problem 29

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
\begin{array}{l}{f(x)=x^{3}+1} \\ {\text { a. }[2,3] \quad \text { b. }[-1,1]}\end{array}
$$

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Problem 30

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
\begin{array}{l}{g(x)=x^{2}} \\ {\text { a. }[-1,1] \quad \text { b. }[-2,0]}\end{array}
$$

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Problem 31

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
\begin{array}{l}{h(t)=\cot t} \\ {\text { a. }[\pi / 4,3 \pi / 4] \quad \text { b. }[\pi / 6, \pi / 2]}\end{array}
$$

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Problem 32

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
\begin{array}{l}{g(t)=2+\cos t} \\ {\text { a. }[0, \pi] \quad \text { b. }[-\pi, \pi]}\end{array}
$$

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Problem 33

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]
$$

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Problem 34

In Exercises 29–34, find the average rate of change of the function over the given interval or intervals.
$$
P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]
$$

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Problem 35

A Ford Mustang Cobra's speed The accompanying figure shows the time-to-distance graph for a 1994 Ford Mustang Cobra accelerating from a standstill.
a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4},$ arranging them in order in a table like the one in Figure $2.3 .$ What are the appropriate units for these slopes?
b. Then estimate the Cobra's speed at time $t=20 \mathrm{sec}$ .

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Problem 36

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 $\mathrm{m}$ to the surface of the moon.
a. Estimate the slopes of the secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4},$ arranging them in a table like the one in Figure $2.3 .$
b. About how fast was the object going when it hit the surface?

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Problem 37

The profits of a small company for each of the first five years of its operation are given in the following table:
a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can. b. What is the average rate of increase of the profits between 1992 and 1994 ?
c. Use your graph to estimate the rate at which the profits were changing in 1992 .

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Problem 38

Make a table of values for the function $F(x)=(x+2) /(x-2)$ at the points $x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000$ $x=10001 / 10000,$ and $x=1 .$
a. Find the average rate of change of $F(x)$ over the intervals $[1, x]$ for each $x \neq 1$ in your table.
b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1$ .

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Problem 39

Let $g(x)=\sqrt{x}$ for $x \geq 0$ .
a. Find the average rate of change of $g(x)$ with respect to $x$ over the intervals $[1,2],[1,1.5]$ and $[1,1+h] .$
b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ $\quad$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$
and $0.000001 .$
c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$
d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$

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Problem 40

Let $f(t)=1 / t$ for $t \neq 0$
a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T .$
b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T],$ for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001$ and 2.000001
c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$

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Problem 41

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}
$$

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Problem 42

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow-1} \frac{x^{3}-x^{2}-5 x-3}{(x+1)^{2}}
$$

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Problem 43

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-1}{x}
$$

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Problem 44

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x^{2}+7}-4}
$$

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Problem 45

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}
$$

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Problem 46

In Exercises 41–46, use a CAS to perform the following steps:
a. Plot the function near the point being approached.
b. From your plot guess the value of the limit.
$$
\lim _{x \rightarrow 0} \frac{2 x^{2}}{3-3 \cos x}
$$

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