### 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 $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists.

You can use the quotient rule to evaluate $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}.$

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

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

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

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

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

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

$\underset{x\to \pi \text{/}2}{\text{lim}}\frac{\text{cot}\phantom{\rule{0.1em}{0ex}}x}{\text{cos}\phantom{\rule{0.1em}{0ex}}x}$

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

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

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

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

$\underset{x\to 0}{\text{lim}}{x}^{2}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(2\pi x\right)=0$

$\underset{x\to 0}{\text{lim}}{x}^{3}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(\frac{\pi}{x}\right)=0$

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

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

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

$f\left(x\right)=\{\begin{array}{l}\sqrt{x+1},x>\text{\u2212}1\\ {x}^{2}+c,x\le -1\end{array}$

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

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

A ball is thrown into the air and the vertical position is given by $x\left(t\right)=\mathrm{-4.9}{t}^{2}+25t+5.$ 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 $x\left(t\right)={t}^{2}-2t+4,$ where *x* is measured in meters and *t* is measured in seconds. Find the average velocity over the time period $t=\left[0,2\right].$

From the previous exercises, estimate the instantaneous velocity at $t=2$ by checking the average velocity within $t=0.01\phantom{\rule{0.2em}{0ex}}\text{sec}\text{.}$