Limits and Continuity

1.1 Assess Your Understanding

Concepts and Vocabulary

Question

Multiple Choice The limit as $x$ approaches $c$ of a function $f$ is written symbolically as [(a) $\lim f(x)$, (b) $\lim\limits_{c \to x} f (x)$, (c) $\lim\limits_{x\to c} f(x)$]

Question

True or False The tangent line to the graph of $f$ at a point $P=(c,f(c))$ is the limiting position of the secant lines passing through $P$ and a point $(x,f(x))$, $x\neq c,$ as $x$ moves closer to $c$.

Question

True or False If $f$ is not defined at $x=c$, then $\lim\limits_{x\rightarrow c}f(x)$ does not exist.

Question

True or False The limit $L$ of a function $y=f(x)$ as $x$ approaches the number $c$ depends on the value of $f$ at $c$.

Question

If $\lim\limits_{x\rightarrow c}\dfrac{f(x)-f(c)}{x-c}$ exists, it equals the ______ of the tangent line to the graph of $f$ at the point $(c,f(c))$.

Question

True or False The limit of a function $y=f(x)$ as $x$ approaches a number $c$ equals $L$ if at least one of the one-sided limits as $x$ approaches $c$ equals $L$.

Skill Building

In Problems 7-12, complete each table and investigate the limit.

Question

$\lim\limits_{x\rightarrow 1}2x$

$\underrightarrow{x~\hbox{approaches 1 from the left}}$								$\underleftarrow{x~\hbox{approaches 1 from the right}}$
$x$	$0.9$	$0.09$	$0.999$	$\rightarrow$	$1$	$\leftarrow$	$1.001$	$1.01$	$1.1$
$f(x)=2x$

Question

$\lim\limits_{x\rightarrow 2} (x+3)$

$\underrightarrow{x~\hbox{approaches 2 from the left}}$								$\underleftarrow{x~\hbox{approaches 2 from the right}}$
$x$	$1.9$	$1.99 $	$1.999$	$\rightarrow$	2	$\leftarrow $	$2.001$	$2.01$	$2.1$
$f(x)=x+3$

Question

$\lim\limits_{x\rightarrow 0}( x^{2}+2)$

$\underrightarrow{x~\hbox{approaches 0 from the left}}$								$\underleftarrow{x~\hbox{approaches 0 from the right}}$
$x$	$-0.1$	$-0.01$	$-0.001$	$\rightarrow $	$0$	$\leftarrow $	$0.001$	$0.01$	0.1
$f(x)=x^{2}+2$

Question

$\lim\limits_{x\rightarrow -1} ( x^{2}-2)$

$\underrightarrow{x~\hbox{approaches –1 from the left}}$								$\underleftarrow{x~\hbox{approaches –1 from the right}}$
$x$	$-1.1$	$-1.01$	$-1.001$	$\rightarrow $	$-1$	$\leftarrow $	$-0.999$	$-0.99$	$-0.9$
$f(x)=x^{2}-2$

Question

$\lim\limits_{x\rightarrow -3}\dfrac{x^{2}-9}{x+3}$

$\underrightarrow{x~\hbox{approaches –3 from the left}}$								$\underleftarrow{x~\hbox{approaches –3 from the right}}$
$x$	$-3.5$	$-3.1$	$-3.01$	$\rightarrow $	$-3$	$\leftarrow $	$-2.99$	$-2.9$	$-2.5$
$f(x)=\dfrac{x^{2}-9}{x+3}$

Question

$\lim\limits_{x\rightarrow -1}\dfrac{x^{3}+1}{x+1}$

$\underrightarrow{x~\hbox{approaches –1 from the left}}$								$\underleftarrow{x~\hbox{approaches –1 from the right}}$
$x$	$-1.1$	$-1.01$	$-1.001$	$\rightarrow $	$-1$	$\leftarrow $	$-0.999$	$-0.99$	$-0.9$
$f(x)=\dfrac{x^{3}+1}{x+1}$

In Problems 13-16, use technology to complete the table and investigate the limit.

Question

$\lim\limits_{x\rightarrow 0}\dfrac{2-2e^{x}}{x}$

$\underrightarrow{x~\hbox{approaches 0 from the left}}$								$\underleftarrow{x~\hbox{approaches 0 from the right}}$
$x$	$-0.2$	$-0.1$	$-0.01$	$\rightarrow $	0	$\leftarrow $	$0.01$	0.1	$0.2$
$f(x)=\dfrac{2-2e^{x}}{x}$

Question

$\lim\limits_{x\rightarrow 1}\dfrac{\ln x}{x-1}$

$\underrightarrow{x~\hbox{approaches 1 from the left}}$								$\underleftarrow{x~\hbox{approaches 1 from the right}}$
$x$	$0.9$	$0.99$	$0.999$	$\rightarrow $	$1$	$\leftarrow $	$1.001$	$1.01$	$1.1$
$f(x)=\dfrac{\ln x}{x-1}$

Question

$\lim\limits_{x\rightarrow 0}\dfrac{1-\cos x}{x}$, where $x$ is measured in radians

$\underrightarrow{x~\hbox{approaches 0 from the left}}$								$\underleftarrow{x~\hbox{approaches 0 from the right}}$
$x$ (in radians)	$-0.2$	$-0.1$	$-0.01$	$\rightarrow $	$0$	$\leftarrow $	$0.01$	0.1	$0.2$
$f(x)=\dfrac{1-\cos x}{x}$

Question

$\lim\limits_{x\rightarrow 0}\dfrac{\sin x}{1+\tan x}$, where $x$ is measured in radians

$\underrightarrow{x~\hbox{approaches 0 from the left}}$								$\underleftarrow{x~\hbox{approaches 0 from the right}}$
$x$ (in radians)	$-0.2$	$-0.1$	$-0.01$	$\rightarrow $	$0$	$\leftarrow $	$0.01$	0.1	$0.2$
$f(x)=\dfrac{\sin x}{1+\tan x}$

In Problems 17—20, use the graph to investigate

$\lim\limits_{x\rightarrow 2^{-}}f(x)$,
$\lim\limits_{x\rightarrow 2^{+}}f(x)$,
$\lim\limits_{x\rightarrow 2}f(x)$.

Question

In Problems 21—28, use the graph to investigate $\lim\limits_{x\rightarrow c}f(x)$. If the limit does not exist, explain why.

Question

In Problems 29—36, use a graph to investigate $\lim\limits_{x\rightarrow c} f(x)$ at the number $c.$

Question

$f(x)=\left\{ \begin{array}{l@{\quad}rl} 2x+5 & \hbox{if} & x\leq 2 \\[1pt] 4x+1 & \hbox{if} & x\gt 2 \end{array} \right.$ at $c=2$

Question

$f(x)=\left\{ \begin{array}{c@{\quad}rl} 2x+1 & \hbox{if} & x\leq 0 \\[1pt] 2x & \hbox{if} & x\gt 0 \end{array} \right.$ at $c=0$

Question

$f(x)=\left\{ \begin{array}{c@{\quad}rl} 3x-1 & \hbox{if} & x\lt 1 \\[1pt] 4 & \hbox{if} & x=1 \\[1pt] 4x & \hbox{if} & x\gt 1 \end{array} \right.$ at $c=1$

Question

$f(x)=\left\{ \begin{array}{c@{\quad}rl} x+2 & \hbox{if} & x\lt 2 \\[1pt] 4 & \hbox{if} & x=2 \\[1pt] x^{2} & \hbox{if} & x\gt 2 \end{array} \right.$ at $c=2$

Question

$f(x)=\left\{ \begin{array}{c@{\quad}rl} 2x^{2} & \hbox{if} & x\lt 1 \\[1pt] 3x^{2}-1 & \hbox{if} & x\gt 1 \end{array} \right.$ at $c=1$

Question

$f(x)=\left\{ \begin{array}{c@{}rl} x^{3} & \hbox{if} & x\lt -1 \\[1pt] x^{2}-1 & \hbox{if} & x>-1 \end{array} \right.$ at $c=-1$

Question

$f(x)=\left\{ \begin{array}{c@{}rl} x^{2} & \hbox{if} & x\leq 0 \\[1pt] 2x+1 & \hbox{if} & x>0 \end{array} \right. $ at $c=0$

Question

$f(x)=\left\{ \begin{array}{c@{}rl} x^{2} & \hbox{if} & x\lt 1 \\[1pt] 2 & \hbox{if} & x=1 \\[1pt] -3x+2 & \hbox{if} & x>1 \end{array} \right.$ at $c=1$

Applications and Extensions

In Problems 37—40, sketch a graph of a function with the given properties. Answers will vary.

Question

$\lim\limits_{x\rightarrow 2}f(x) =3;$ $\lim\limits_{x\rightarrow 3^{-}}f(x) =3;$ $\lim\limits_{x\rightarrow 3^{+}}f(x) =1;$ $f(2) =3;$ $f(3) =1$

Question

$\lim\limits_{x\rightarrow -1}f(x) =0;$ $ \lim\limits_{x\rightarrow 2^{-}}f(x) =-2;$ $ \lim\limits_{x\rightarrow 2^{+}}f(x) =-2;$ $ f(-1)$ is not defined; $f(2) =-2$

Question

$\lim\limits_{x\rightarrow 1}f(x) =4;$ $ \lim\limits_{x\rightarrow 0^{-}}f(x) =-1;$ $ \lim\limits_{x\rightarrow 0^{+}}f(x) =0;$ $f(0) =-1;$ $f(1) =2$

Question

$\lim\limits_{x\rightarrow 2}f(x) =2; \lim\limits_{x\rightarrow -1}f(x) =0;$ $\lim\limits_{x\rightarrow 1}f(x) =1;$ $f(-1) =1;$ $f(2) =3$

In Problems 41—50, use either a graph or a table to investigate each limit.

Question

$\lim\limits_{x\rightarrow 5^{+}}\dfrac{|x-5|}{x-5}$

Question

$\lim\limits_{x\rightarrow 5^{-}}\dfrac{|x-5|}{x-5}$

Question

$\lim\limits_{x\rightarrow \left(\frac{1}{2}\right) ^{-}}\lfloor 2x\rfloor$

Question

${\lim\limits_{x\rightarrow \left(\frac{1}{2}\right) ^{+}}}\lfloor 2x\rfloor$

Question

${\lim\limits_{x\rightarrow \left(\frac{2}{3}\right) ^{-}}}\lfloor 2x\rfloor$

Question

${\lim\limits_{x\rightarrow \left(\frac{2}{3}\right) ^{+}}}\lfloor 2x\rfloor$

Question

${\lim\limits_{x\rightarrow 2^{+}}}\ \sqrt{{|}x{|}-x}$

Question

${\lim\limits_{x\rightarrow 2^{-}}}\ \sqrt{{|}x{|}-x}$

Question

${\lim\limits_{x\rightarrow 2^{+}}}\ \sqrt[3]{\lfloor x\rfloor -x}$

Question

${\lim\limits_{x\rightarrow 2^{-}}}\ \sqrt[3]{\lfloor x\rfloor -x}$

Question

Slope of a Tangent Line For $f(x)=3x^{2}$:

Find the slope of the secant line containing the points $(2,12)$ and $(3,27)$.
Find the slope of the secant line containing the points $(2,12)$ and $(x,f(x))$, $x\neq 2$.
Create a table to investigate the slope of the tangent line to the graph of $f$ at $2$ using the result from (b).
On the same set of axes, graph $f$, the tangent line to the graph of $f$ at the point $(2,12)$, and the secant line from (a).

Question

Slope of a Tangent Line For $f(x)=x^{3}$:

Find the slope of the secant line containing the points $(2,8)$ and $(3,27)$.
Find the slope of the secant line containing the points $(2,8)$ and $(x,f(x)),$ $x\neq 2$.
Create a table to investigate the slope of the tangent line to the graph of $f$ at $2$ using the result from (b).
On the same set of axes, graph $f$, the tangent line to the graph of $f$ at the point $(2, 8)$, and the secant line from (a).

Question

Slope of a Tangent Line For $f(x)=\dfrac{1}{2}x^{2}-1$:

Find the slope $m_{\mathrm{sec}}$ of the secant line containing the points $P=(2,f(2))$ and $Q=(2+h,f(2+h))$.
Use the result from (a) to complete the following table:

h -0.5 -0.1 -0.001 0.001 0.1 0.5

$m_{\sec }$
Investigate the limit of the slope of the secant line found in (a) as $h\rightarrow 0$.
What is the slope of the tangent line to the graph of $f$ at the point $P=(2,f(2))$?
On the same set of axes, graph $f$ and the tangent line to $f$ at $P=(2,f(2))$.

Question

Slope of a Tangent Line For $f(x)=x^{2}-1$:

Find the slope $m_{\mathrm{sec}}$ of the secant line containing the points $P=(-1,f(-1))$ and $Q=(-1+h,f(-1+h))$.
Use the result from (a) to complete the following table:

h -0.1 -0.01 -0.001 -0.0001 0.0001 0.001 0.01 0.1

m_sec
Investigate the limit of the slope of the secant line found in (a) as $h\rightarrow 0$.
What is the slope of the tangent line to the graph of $f$ at the point $P=(-1,f(-1) )$?
On the same set of axes, graph $f$ and the tangent line to $f$ at $P=(-1,f(-1))$.

Question

Investigate $\lim\limits_{x\rightarrow 0}\cos \dfrac{\pi}{x}$ by using a table and evaluating the function $f(x) =\cos \dfrac{\pi }{x}$ at $x$ =$-\dfrac{1}{2}, -\dfrac{1}{4}, - \dfrac{1}{8}, -\dfrac{1}{10}$, $-\dfrac{1}{12},\ldots, \dfrac{1}{12}, \dfrac{1}{10}, \dfrac{1}{8}, \dfrac{1}{4}, \dfrac{1}{2}$.
Investigate $\lim\limits_{x\rightarrow 0}\cos \dfrac{\pi }{x}$ by using a table and evaluating the function $f(x) =\cos \dfrac{\pi }{x}$ at $x=-1,-\dfrac{1}{3}, -\dfrac{1}{5}, -\dfrac{1}{7}, - \dfrac{1}{9},\ldots,\dfrac{1}{9},\dfrac{1}{7}, \dfrac{1}{5}, \dfrac{1}{3},1$.
Compare the results from (a) and (b). What do you conclude about the limit? Why do you think this happens? What is your view about using a table to draw a conclusion about limits?
Use graphing technology to graph $f$. Begin with the $x$ -window $[-2\pi ,2\pi ]$ and the $y$-window [-1,1]. If you were finding $\lim\limits_{x\rightarrow 0}f(x)$ using a graph, what would you conclude? Zoom in on the graph. Describe what you see. (${Hint}$: Be sure your calculator is set to the radian mode.)

Question

Investigate $\lim\limits_{x\rightarrow 0}\cos \dfrac{ \pi }{x^{2}}$ by using a table and evaluating the function $f(x) =\cos \dfrac{\pi }{x^{2}}$ at $x=-0.1$, $-0.01$, $-0.001$, $-0.0001$, $ 0.0001$, $0.001$, $0.01$, $0.1$.
Investigate $\lim\limits_{x\rightarrow 0}\cos \dfrac{\pi }{x^{2} }$ by using a table and evaluating the function $f(x) =\cos \dfrac{\pi }{x^{2}}$ at $x=-\dfrac{2}{3}$, $-\dfrac{2}{5}$, $-\dfrac{2}{7}$, $-\dfrac{2}{9}, \ldots, \dfrac{2}{9},\dfrac{2}{7}$, $\dfrac{2}{5}$, $\dfrac{2 }{3}$.
Compare the results from (a) and (b). What do you conclude about the limit? Why do you think this happens? What is your view about using a table to draw a conclusion about limits?
Use graphing technology to graph $f$. Begin with the $x$ -window $\left[ -2\pi ,2\pi \right]$ and the $y$-window $[-1,1]$. If you were finding $\lim\limits_{x\rightarrow 0}f(x)$ using a graph, what would you conclude? Zoom in on the graph. Describe what you see. (${Hint}$: Be sure your calculator is set to the radian mode.)

Question

Use a table to investigate $\lim\limits_{x\rightarrow 2}\dfrac{x-8}{2}$.
How close must $x$ be to $2,$ so that $f(x)$ is within $0.1$ of the limit?
80
How close must $x$ be to $2,$ so that $f(x)$ is within $0.01$ of the limit?

Question

Use a table to investigate $\lim\limits_{x\rightarrow 2} (5-2x)$.
How close must $x$ be to $2,$ so that $f(x)$ is within $0.1$ of the limit?
How close must $x$ be to $2,$ so that $f(x)$ is within $0.01$ of the limit?

Question

First-Class Mail As of January 2013, the U.S. Postal Service charged $0.46 postage for first-class letters weighing up to and including 1 ounce, plus a flat fee of $0.20 for each additional or partial ounce up to and including 3.5 ounces. First-class letter rates do not apply to letters weighing more than 3.5 ounces.

Source: U.S. Postal Service Notice 123.

Find a function $C$ that models the first-class postage charged, in dollars, for a letter weighing $w$ ounces. Assume $w>0$.
What is the domain of $C$?
Graph the function $C.$
Use the graph to investigate $\lim\limits_{w\rightarrow 2^{-}}C(w)$ and $\lim\limits_{w\rightarrow 2^{+}}C(w).$ Do these suggest that $\lim\limits_{w\rightarrow 2}$ $C(w)$ exists?
Use the graph to investigate $\lim\limits_{w\rightarrow 0^{+}}C(w)$.
Use the graph to investigate $\lim\limits_{w\rightarrow 3.5^{-}}C(w)$.

Question

First-Class Mail As of January 2013, the U.S. Postal Service charged $0.92 postage for first-class retail flats (large envelopes) weighing up to and including 1 ounce, plus a flat fee of $0.20 for each additional or partial ounce up to and including 13 ounces. First-class rates do not apply to flats weighing more than 13 ounces.

Find a function $C$ that models the first-class postage charged, in dollars, for a large envelope weighing $w$ ounces. Assume $w\gt 0$.
What is the domain of $C?$
Graph the function $C.$
Use the graph to investigate $\lim\limits_{w\rightarrow 1^{-}}C(w)$ and $\lim\limits_{w\rightarrow 1^{+}}C(w).$ Do these suggest that $\lim\limits_{w\rightarrow 1}$ $C(w)$ exists?
Use the graph to investigate $\lim\limits_{w\rightarrow 12^{-}}C(w)$ and $\lim\limits_{w\rightarrow 12^{+}}C(w)$. Do these suggest that $\lim\limits_{w\rightarrow 12}$ $C(w)$ exists?
Use the graph to investigate $\lim\limits_{w\rightarrow 0^{+}}C(w).$
Use the graph to investigate $\lim\limits_{w\rightarrow 13^{-}}C(w).$

Source: U.S. Postal Service Notice 123.

Question

Correlating Student Success to Study Time Professor Smith claims that a student's final exam score is a function of the time $t$ (in hours) that the student studies. He claims that the closer to seven hours one studies, the closer to 100% the student scores on the final. He claims that studying significantly less than seven hours may cause one to be underprepared for the test, while studying significantly more than seven hours may cause ''burnout.''

Write Professor Smith's claim symbolically as a limit.
Write Professor Smith's claim using the $\epsilon-\delta$ definition of limit.

Source: Submitted by the students of Millikin University.

Question

The definition of the slope of the tangent line to the graph of $y=f(x)$ at the point $(c,f(c))$ is $m_{\tan }=\lim\limits_{x\rightarrow c}\dfrac{f(x) -f(c) }{x-c}$. Another way to express this slope is to define a new variable $h=x-c$. Rewrite the slope of the tangent line $m_{\tan }$ using $h$ and $c$.

Question

If $f(2) =6,$ can you conclude anything about $\lim\limits_{x\rightarrow 2}f(x) ?$ Explain your reasoning.

Question

If $\lim\limits_{x\rightarrow 2}f(x) =6,$ can you conclude anything about $f(2) ?$ Explain your reasoning.

Question

The graph of $f(x) ={\dfrac{{x-3}}{{3-x}}}$ is a straight line with a point punched out.

What straight line and what point?
Use the graph of $f$ to investigate the one-sided limits of $f$ as $x$ approaches $3.$
Does the graph suggest that $\lim\limits_{x\rightarrow 3}f(x) $ exists? If so, what is it?

Question

Use a table to investigate $\lim\limits_{x \rightarrow 0}(1+x) ^{1/x}.$
Use graphing technology to graph $g(x) =(1+x) ^{1/x}$.
What do (a) and (b) suggest about $\lim\limits_{x\rightarrow 0}(1+x) ^{1/x}?$
Find $\lim\limits_{x\rightarrow 0}(1+x) ^{1/x}.$

Challenge Problems

For Problems 67—70, investigate each of the following limits. \[ f(x) ={\left\{ \begin{array}{c@{\qquad}l} 1 & \hbox{if }x\hbox{ is an integer} \\ 0 & \hbox{if }x\hbox{ is not an integer} \end{array} \right. } \]

Question

$\lim\limits_{x\rightarrow 2}f(x)$

Question

$\lim\limits_{x\rightarrow 1/2}f(x)$

Question

$\lim\limits_{x\rightarrow 3}f(x)$

Question

$\lim\limits_{x\rightarrow 0}f(x)$

\(\underrightarrow{x~\hbox{approaches 1 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 1 from the right}}\)
\(x\)	\(0.9\)	\(0.09\)	\(0.999\)	\(\rightarrow\)	\(1\)	\(\leftarrow\)	\(1.001\)	\(1.01\)	\(1.1\)
\(f(x)=2x\)

\(\underrightarrow{x~\hbox{approaches 2 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 2 from the right}}\)
\(x\)	\(1.9\)	\(1.99 \)	\(1.999\)	\(\rightarrow\)	2	\(\leftarrow \)	\(2.001\)	\(2.01\)	\(2.1\)
\(f(x)=x+3\)

\(\underrightarrow{x~\hbox{approaches 0 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\)	\(-0.1\)	\(-0.01\)	\(-0.001\)	\(\rightarrow \)	\(0\)	\(\leftarrow \)	\(0.001\)	\(0.01\)	0.1
\(f(x)=x^{2}+2\)

\(\underrightarrow{x~\hbox{approaches –1 from the left}}\)								\(\underleftarrow{x~\hbox{approaches –1 from the right}}\)
\(x\)	\(-1.1\)	\(-1.01\)	\(-1.001\)	\(\rightarrow \)	\(-1\)	\(\leftarrow \)	\(-0.999\)	\(-0.99\)	\(-0.9\)
\(f(x)=x^{2}-2\)

\(\underrightarrow{x~\hbox{approaches –3 from the left}}\)								\(\underleftarrow{x~\hbox{approaches –3 from the right}}\)
\(x\)	\(-3.5\)	\(-3.1\)	\(-3.01\)	\(\rightarrow \)	\(-3\)	\(\leftarrow \)	\(-2.99\)	\(-2.9\)	\(-2.5\)
\(f(x)=\dfrac{x^{2}-9}{x+3}\)

\(\underrightarrow{x~\hbox{approaches 0 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\)	\(-0.2\)	\(-0.1\)	\(-0.01\)	\(\rightarrow \)	0	\(\leftarrow \)	\(0.01\)	0.1	\(0.2\)
\(f(x)=\dfrac{2-2e^{x}}{x}\)

\(\underrightarrow{x~\hbox{approaches 1 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 1 from the right}}\)
\(x\)	\(0.9\)	\(0.99\)	\(0.999\)	\(\rightarrow \)	\(1\)	\(\leftarrow \)	\(1.001\)	\(1.01\)	\(1.1\)
\(f(x)=\dfrac{\ln x}{x-1}\)

\(\underrightarrow{x~\hbox{approaches 0 from the left}}\)								\(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\) (in radians)	\(-0.2\)	\(-0.1\)	\(-0.01\)	\(\rightarrow \)	\(0\)	\(\leftarrow \)	\(0.01\)	0.1	\(0.2\)
\(f(x)=\dfrac{1-\cos x}{x}\)

h	-0.5	-0.1	-0.001	0.001	0.1	0.5
\(m_{\sec }\)

h	-0.1	-0.01	-0.001	-0.0001	0.0001	0.001	0.01	0.1
m_sec