### Practice Test

For the following exercises, use the graph of $f$ in Figure 1.

$\underset{x\to {\mathrm{-1}}^{+}}{\mathrm{lim}}f(x)$

$\underset{x\to \mathrm{-1}}{\mathrm{lim}}f(x)$

At what values of $x$ is $f$ discontinuous? What property of continuity is violated?

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a.$ If the function has a limit as $x$ approaches $a,$ state it. If not, discuss why there is no limit

$f(x)=\{\begin{array}{ll}\frac{1}{x}-3,\text{i}f\hfill & x\le 2\hfill \\ {x}^{3}+1,if\hfill & x2\hfill \end{array}\text{}a=2$

$f(x)=\{\begin{array}{lll}{x}^{3}+1,\hfill & if\hfill & x<1\hfill \\ 3{x}^{2}-1,\hfill & if\hfill & x=1\hfill \\ -\sqrt{x+3}+4,\hfill & if\hfill & x>1\hfill \end{array}\text{}a=1$

For the following exercises, evaluate each limit using algebraic techniques.

$\underset{x\to \mathrm{-5}}{\mathrm{lim}}\left(\frac{\frac{1}{5}+\frac{1}{x}}{10+2x}\right)$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{1}{h}-\frac{1}{{h}^{2}+h}\right)$

For the following exercises, determine whether or not the given function $f$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

$f(x)=\frac{{x}^{3}-4{x}^{2}-9x+36}{{x}^{3}-3{x}^{2}+2x-6}$

For the following exercises, use the definition of a derivative to find the derivative of the given function at $x=a.$

$f(x)=\frac{3}{\sqrt{x}}$

For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

$f(x)=\frac{2}{1+{e}^{\frac{2}{x}}}$

For the following exercises, explain the notation in words when the height of a projectile in feet, $s,$ is a function of time $t$ in seconds after launch and is given by the function $s(t).$

$s(2)$

$\frac{s(2)-s(1)}{2-1}$

For the following exercises, use technology to evaluate the limit.

$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}(x)}{3x}$

$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}(x)(1-\mathrm{cos}(x))}{2{x}^{2}}$

Evaluate the limit by hand.

$\begin{array}{c}\mathrm{lim}\\ {}^{x\to 1}\end{array}f(x),\text{where}f(x)=\{\begin{array}{cc}4x-7& x\ne 1\\ {x}^{2}-4& x=1\end{array}$

At what value(s) of $x$ is the function below discontinuous?

$f(x)=\{\begin{array}{c}4x-7\phantom{\rule{0.5em}{0ex}}x\ne 1\\ {x}^{2}-4\phantom{\rule{0.5em}{0ex}}x=1\end{array}$

For the following exercises, consider the function whose graph appears in Figure 3.

Find all values of $x$ at which $f\text{'}(x)$ does not exist.

Find an equation of the tangent line to the graph of $f$ the indicated point: $f(x)=3{x}^{2}-2x-6,\text{}x=-2$

For the following exercises, use the function $f(x)=x{\left(1-x\right)}^{\frac{2}{5}}$.

Graph the function $f(x)=x{\left(1-x\right)}^{\frac{2}{5}}$ by entering $f(x)=x{\left({\left(1-x\right)}^{2}\right)}^{\frac{1}{5}}$ and then by entering $f(x)=x{\left({\left(1-x\right)}^{\frac{1}{5}}\right)}^{2}$.

Explore the behavior of the graph of $f(x)$ around $x=1$ by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at $x=1.$

For the following exercises, find the derivative of each of the functions using the definition: $\underset{h\to 0}{\mathrm{lim}}\frac{f(x+h)-f(x)}{h}$

$f(x)=2x-8$

$f(x)=x-\frac{1}{2}{x}^{2}$

$f(x)=\frac{3}{x-1}$

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