Chapter Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

A function has to be continuous at $x=a$ if the $\underset{x\to a}{\text{lim}}f(x)$ exists.

You can use the quotient rule to evaluate $\underset{x\to 0}{\text{lim}}\frac{ \sin x}{x}.$

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If there is a vertical asymptote at $x=a$ for the function $f(x),$ then $f$ is undefined at the point $x=a.$

If $\underset{x\to a}{\text{lim}}f(x)$ does not exist, then $f$ is undefined at the point $x=a.$

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Using the graph, find each limit or explain why the limit does not exist.

$\underset{x\to -1}{\text{lim}}f(x)$
$\underset{x\to 1}{\text{lim}}f(x)$
$\underset{x\to {0}^{+}}{\text{lim}}f(x)$
$\underset{x\to 2}{\text{lim}}f(x)$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$\underset{x\to 2}{\text{lim}}\frac{2{x}^{2}-3x-2}{x-2}$

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$\underset{x\to 0}{\text{lim}}3{x}^{2}-2x+4$

$\underset{x\to 3}{\text{lim}}\frac{{x}^{3}-2{x}^{2}-1}{3x-2}$

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$8\text{/}7$

$\underset{x\to \pi \text{/}2}{\text{lim}}\frac{ \cot x}{ \cos x}$

$\underset{x\to -5}{\text{lim}}\frac{{x}^{2}+25}{x+5}$

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$\underset{x\to 2}{\text{lim}}\frac{3{x}^{2}-2x-8}{{x}^{2}-4}$

$\underset{x\to 1}{\text{lim}}\frac{{x}^{2}-1}{{x}^{3}-1}$

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$2\text{/}3$

$\underset{x\to 1}{\text{lim}}\frac{{x}^{2}-1}{\sqrt{x}-1}$

$\underset{x\to 4}{\text{lim}}\frac{4-x}{\sqrt{x}-2}$

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$\underset{x\to 4}{\text{lim}}\frac{1}{\sqrt{x}-2}$

In the following exercises, use the squeeze theorem to prove the limit.

$\underset{x\to 0}{\text{lim}}{x}^{2} \cos (2\pi x)=0$

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Since $-1\le \cos (2\pi x)\le 1,$ then $-{x}^{2}\le {x}^{2} \cos (2\pi x)\le {x}^{2}.$ Since $\underset{x\to 0}{\text{lim}}{x}^{2}=0=\underset{x\to 0}{\text{lim}}-{x}^{2},$ it follows that $\underset{x\to 0}{\text{lim}}{x}^{2} \cos (2\pi x)=0.$

$\underset{x\to 0}{\text{lim}}{x}^{3} \sin (\frac{\pi }{x})=0$

Determine the domain such that the function $f(x)=\sqrt{x-2}+x{e}^{x}$ is continuous over its domain.

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$\left[2,\infty \right]$

In the following exercises, determine the value of $c$ such that the function remains continuous. Draw your resulting function to ensure it is continuous.

$f(x)=\bigg\{\begin{array}{l}{x}^{2}+1,x>c\\ 2x,x\le c\end{array}$

$f(x)=\bigg\{\begin{array}{l}\sqrt{x+1},x>\text{−}1\\ {x}^{2}+c,x\le -1\end{array}$

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$c=-1$

In the following exercises, use the precise definition of limit to prove the limit.

$\underset{x\to 1}{\text{lim}}(8x+16)=24$

$\underset{x\to 0}{\text{lim}}{x}^{3}=0$

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$\delta =\sqrt[3]{\epsilon }$

A ball is thrown into the air and the vertical position is given by $x(t)=-4.9{t}^{2}+25t+5.$ Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

A particle moving along a line has a displacement according to the function $x(t)={t}^{2}-2t+4,$ where $x$ is measured in meters and $t$ is measured in seconds. Find the average velocity over the time period $t=\left[0,2\right].$

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$0\text{m}\text{/} \sec$

From the previous exercises, estimate the instantaneous velocity at $t=2$ by checking the average velocity within $t=0.01 \sec \text{.}$

2. Limits

Chapter 2 Review Exercises

Chapter Review Exercises