Limits and Continuity

REVIEW EXERCISES
Printed Page 9999

Use a table of numbers to investigate ${\lim\limits_{x \rightarrow 0}}\dfrac{1-\cos x}{1+ \cos x}$ .

The table suggests $\mathop {\lim\limits_{x \to 0}} \dfrac{1 - \cos x}{1 + \cos x} = 0$ .

	$\xrightarrow{x\ {\rm approaches}\ 0\atop \mbox{from the left}}$				$\xleftarrow{x\ {\rm approaches}\ 0\atop \mbox{from the right}}$
$x$	$-0.1$	$-0.01$	$-0.001$	$\rightarrow 0 \leftarrow$	$0.001$	$0.01$	$0.1$
$f(x)=\dfrac{1 - \cos x}{1 + \cos x}$	0.002504	0.000025	0.00000025	$f(x)$ approaches $0$	0.00000025	0.000025	0.002504

In Problems 2 and 3, use a graph to investigate $\lim\limits_{x\rightarrow c}f(x)$ .

$f(x)=\left\{ \begin{array}{l@{\quad}l@{\quad}l} 2x-5 & & \hbox{if }x\lt1 \\ 6-9x & & \hbox{if }x\geq 1 \end{array} \right.$ $\quad$ at $c=1$

$f(x)=\left\{ \begin{array}{l@{\quad}l@{\quad}l} x^{2}+2 & & \hbox{if }x\lt2 \\ 2x+1 & & \hbox{if }x\geq 2 \end{array} \right.$ $\quad$ at $c=2$

$\mathop {\lim\limits_{x \to 2}} f(x)$ does not exist.

For $f(x)=x^{2}-3$ :
1. Find the slope of the secant line joining $(1,-2)$ and $(2,1)$ .
2. Find the slope of the tangent line to the graph of $f$ at $(1,-2)$ .

In Problems 5 and 6, for each function find the limit of the difference quotient $\lim\limits_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}$ .

$f(x)=\dfrac{3}{x}$

$- \dfrac{3}{x^2}$

$f(x)=3x^{2}+2x$

Find $\lim\limits_{x\rightarrow 0}f(x)$ if $1+\sin x\leq f(x)\leq \vert x\vert +1$

In Problems 8-22, find each limit.

$\lim\limits_{x\rightarrow 2}$ $\left( 2x-\dfrac{1}{x}\right)$

$\lim\limits_{x\rightarrow \pi } \left( x\cos x\right)$

$-\pi$

$\lim\limits_{x\rightarrow -1}\left( x^{3}+3x^{2}-x-1\right)$

$\lim\limits_{x\rightarrow 0}\sqrt[3]{x( x+2) ^{3}}$

${\lim\limits_{x\rightarrow 0}}$ $[(2x+3) (x^{5}+5x)]$

${\lim\limits_{x\rightarrow 3}} \dfrac{x^{3}-27}{x-3}$

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

${\lim\limits_{x\rightarrow 2}}\dfrac{x^{2}-4}{x-2}$

${\lim\limits_{x\rightarrow -1}}\dfrac{x^{2}+3x+2} {x^{2}+4x+3}$

${\lim\limits_{x\rightarrow -2}} \dfrac{x^{3}+5x^{2}+6x}{x^{2}+x-2}$

$\dfrac{2}{3}$

${\lim\limits_{x\rightarrow 1}} \left(x^{2}-3x+\dfrac{1}{x}\right) ^{15}$

${\lim\limits_{x\rightarrow 2}}\dfrac{3-\sqrt{x^{2}+5}}{x^{2}-4}$

$- \dfrac{1}{6}$

${\lim\limits_{x\rightarrow 0}} \left\{ \dfrac{1}{x}\left[ \dfrac{1}{(2+x)^{2}}-\dfrac{1}{4}\right] \right\}$

${\lim\limits_{x\rightarrow 0}} \dfrac{\left({x+3}\right) ^{2}-9}{x}$

${\lim\limits_{x\rightarrow 1}} [(x^{3}-3x^{2}+3x-1) (x+1) ^{2}]$

In Problems 23-28, find each one-sided limit, if it exists.

${\lim\limits_{x\rightarrow - 2^{+}}}\dfrac{ x^{2}+5x+6}{x+2}$

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

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

$-1$

${\lim\limits_{x\rightarrow \,3/2^{+}}}\lfloor 2x\rfloor$

${\lim\limits_{x\rightarrow 4^{-}}}\dfrac{x^{2}-16}{x-4}$

${\lim\limits_{x\rightarrow 1^{+}}},\sqrt{x-1}$

In Problems 29 and 30, find $\lim\limits_{x\rightarrow c^{-}}f(x)$ and $\lim\limits_{x\rightarrow c^{+}}f(x)$ for the given $c$ Determine whether $\lim\limits_{x\rightarrow c}f(x)$ exists.

$f(x)=\left\{ \begin{array}{l@{\quad}l} 2x+3 & \hbox{if }x\lt 2 \\[3pt] 9-x & \hbox{if }x\geq 2 \end{array} \right.$ at $c$ =2

$\mathop{\lim\limits_{x \to 2^ - }} f(x) = 7$ , $\mathop{\lim\limits_{x \to 2^ + }} f(x) = 7$ , $\mathop{\lim\limits_{x \to 2}} f(x) = 7$

142

$f(x)=\left\{ \begin{array}{c@{ }c} 3x+1 & \hbox{if }x \lt 3 \\[3pt] 10 & \hbox{if }x=3 \\[3pt] 4x-2 & \hbox{if }x \gt 3 \end{array} \right.$ at $c=3$

In Problems 31-36, determine whether $f$ is continuous at $c$ .

$f(x)=\left\{ \begin{array}{c@{}c} 5x-2 & \hbox{if }x \lt 1 \\[3pt] 5 & \hbox{if }x=1 \\[3pt] 2x+1 & \hbox{if }x \gt 1 \end{array} \right.$ at $c=1$

$f$ is not continuous at $c = 1$ .

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

$f(x)=\left\{ \begin{array}{c@{}l} 4-3x^{2} & \hbox{if }x\lt0 \\[3pt] 4 & \hbox{if }x=0 \\[3pt] \sqrt{16-x^{2}} & \hbox{if }0 \lt x\leq 4 \end{array} \right.$ at $c=0$

$f$ is continuous at $c = 0$ .

$f(x)=\left\{ \begin{array}{c@{}l} \sqrt{4+x} & \hbox{if }-4\leq x\leq 4 \\[3pt] \sqrt{\dfrac{x^{2}-16}{x-4}} & \hbox{if }x>4 \end{array} \right.$ at $c=4$

$f(x) =$ $\lfloor \,2x\rfloor$ at $c=\dfrac{1}{ 2}$

$f$ is not continuous at $c=\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$}$ .

$f(x)=|\,x-5\,|$ at $c=5$

1. Find the average rate of change of $f(x)= 2x^2 -5x$ from 1 to $x$ .
2. Find the limit as $x$ approaches 1 of the average rate of change found in (a).

$2x - 3$
$-1$

A function $f$ is defined on the interval $[-1,1]$ with the following properties: $f$ is continuous on $[-1,1]$ except at $0$ , negative at $-1$ , positive at $1$ , but with no zeros. Does this contradict the Intermediate Value theorem?

In Problems 39-43 find all values $x$ for which $f(x)$ is continuous.

$f(x)=\dfrac{x}{x^{3}-27}$

$f$ is continuous on the set $\{ x \vert x \ne 3 {\}}$ .

$f(x)=\dfrac{x^{2}-3}{x^{2}+5x+6}$

$f(x)=\dfrac{2x+1}{x^{3}+4x^{2}+4x}$

$f$ is continuous on the set $\{ x \vert x \ne -2, x \ne 0 \}$ .

$f(x) =\sqrt{x-1}$

$f(x) =2^{-x}$

$f$ is continuous on the set of all real numbers.

Use the Intermediate Value Theorem to determine whether $2x^{3}+3x^{2}-23x-42=0$ has a zero in the interval $[3,4]$ .

In Problems 45 and 46, use the Intermediate Value Theorem to approximate the zero correct to three decimal places.

$f(x) =8x^{4}-2x^{2}+5x-1$ on the interval $\left[0,1\right]$ .

0.215

$f(x) =3x^{3}-10x+9;$ zero between $-3$ and $-2$ .

Find ${\lim\limits_{x\rightarrow 0^{+}}}\dfrac{|\,x\,|}{x}(1-x)$ and ${\lim\limits_{x\rightarrow 0^{-}}}\dfrac{|\,x\,|}{x}(1-x)$ . What can you say about ${\lim\limits_{x\rightarrow 0}}\dfrac{|x|}{x} (1-x)$ ?

$\mathop {\lim\limits_{x \to 0^ + }} \dfrac{| x |}{x}(1 - x) = 1$ , $\mathop{\lim \limits_{x \to 0^ - }} \dfrac{| x |}{x}(1 - x) = - 1$ , $\mathop{\lim \limits_{x \to 0}} \dfrac{| x |}{x}(1 - x)$ does not exist.

Find ${\lim\limits_{x\rightarrow 2}}\left( \dfrac{x^{2}}{x-2}- \dfrac{2x}{x-2}\right)$ . Then comment on the statement that this limit is given by $\lim\limits_{x\rightarrow 2}\dfrac{x^{2}}{x-2}-\lim\limits_{x\rightarrow 2}\dfrac{2x}{x-2}$ .

Find ${\lim\limits_{h\rightarrow 0}} \dfrac{f(x+h)-f(x)}{h}$ for $f(x)=\sqrt{x}$ .

$\dfrac{1}{2\sqrt x}$

For $\lim\limits_{x\rightarrow 3}(2x+1)=7$ , find the largest possible $\delta$ that “works” for $\epsilon =0.01$ .

In Problems 51-60, find each limit.

$\lim\limits_{x\rightarrow 0}\cos$ (tan x)

$\lim\limits_{x\rightarrow 0}{\dfrac{{\sin {\dfrac{{x}}{{4}}}}}{{x}}}$

$\lim\limits_{x\rightarrow 0}\,\dfrac{\tan (3x) }{\tan ( 4x) }$

$\dfrac{3}{4}$

${\lim\limits_{x\rightarrow 0}}\dfrac{\cos {\dfrac{x}{3}-1}}{x}$

$\lim\limits_{x\rightarrow 0}\left( \dfrac{\cos x-1}{x}\right) ^{10}$

$\lim\limits_{x\rightarrow 0}{\dfrac{{e^{4x}-1}}{e^{x}{-1}}}$

$\lim\limits_{x\rightarrow \pi /2^{+}}\tan x$

$-\infty$

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

$\lim\limits_{x\rightarrow \infty }\dfrac{3x^{3}-2x+1}{x^{3}-8}$

$\lim\limits_{x\rightarrow \infty }\dfrac{3x^{4}+x}{2x^{2}}$

In Problems 61 and 62, find any vertical and horizontal asymptotes of $f$ .

$f(x)=\dfrac{4x-2}{x+3}$

$x = -3$ is a vertical asymptote. $y = 4$ is a horizontal asymptote.

$f(x)=\dfrac{2x}{x^{2}-4}$

Let $f( x) =\left\{ \begin{array}{c@{}c} \dfrac{\tan x}{2x} & \hbox{if }x\neq 0 \\ \dfrac{1}{2} & \hbox{if }x=0 \end{array} \right.$ . Is $f$ continuous at $0$ ?

Yes

Let $f( x) =\left\{ \begin{array}{c@{}c} \dfrac{\sin ( 3x) }{x} & \hbox{if }x\neq 0 \\ 1 & \hbox{if }x=0 \end{array} \right.$ . Is $f$ continuous at $0$ ?

The function $f( x) =\dfrac{\cos \left( \pi x+\dfrac{\pi }{2}\right) }{x}$ is not defined at $0$ . Decide how to define $f( 0)$ so that $f$ is continuous at $0$ .

$f(0) = -\pi$

Use an $\epsilon$ - $\delta$ argument to show that the statement $\lim\limits_{x\rightarrow -3} (x^{2}-9) =-18$ is false.

1. Sketch a graph of a function $f$ that has the following properties: $\begin{eqnarray*} &&\\[-30pt] &&f(-1)=0,\quad \lim_{x\rightarrow \infty }f(x)=2,\quad \lim_{x\rightarrow -\infty }f(x)=2,\\[5pt] &&\lim_{x\rightarrow 4^{-}}f(x)=-\infty, \quad \lim_{x\rightarrow 4^{+}}f(x)=\infty \end{eqnarray*}$

(a) Answers will vary.

1. Find the domain and the intercepts (if any) of $R( x) =\dfrac{2x^{2}-5x+2}{5x^{2}-x-2}$ .
2. Discuss the behavior of the graph of $R$ at numbers where $R$ is not defined.
3. Find any vertical or horizontal asymptotes of the function $R$ .

If $1-x^{2}\leq f(x)\leq$ cos $x$ for all $x$ in the interval $-\dfrac{\pi }{2} \lt x \lt \dfrac{\pi }{2}$ , show that $\lim\limits_{x\rightarrow 0}f(x)=1$ .

See Student Solution Manual.

REVIEW EXERCISESPrinted Page 9999

REVIEW EXERCISES
Printed Page 9999