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For the following exercises, find the limits if $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}f\left(x\right)=\mathrm{-3}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}g\left(x\right)=5.$
$\underset{x\to c}{\mathrm{lim}}\frac{f(x)}{g(x)}$
$\underset{x\to c}{\mathrm{lim}}\left(f(x)\cdot g(x)\right)$
$\mathrm{-15}$
$\underset{x\to {0}^{+}}{\mathrm{lim}}f(x),f(x)=\{\begin{array}{c}3{x}^{2}+2x+1\\ 5x+3\end{array}\text{}\begin{array}{c}x0\\ x0\end{array}$
$\underset{x\to {0}^{-}}{\mathrm{lim}}f(x),f(x)=\{\begin{array}{c}3{x}^{2}+2x+1\\ 5x+3\end{array}\text{}\begin{array}{c}x0\\ x0\end{array}$
3
$\underset{x\to {3}^{+}}{\mathrm{lim}}\left(3x-\mathrm{\u301ax\u301b}\right)$
For the following exercises, evaluate the limits using algebraic techniques.
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(h+6\right)}^{2}-36}{h}\right)$
12
$\underset{x\to 25}{\mathrm{lim}}\left(\frac{{x}^{2}-625}{\sqrt{x}-5}\right)$
$\underset{x\to 1}{\mathrm{lim}}\left(\frac{-{x}^{2}-9x}{x}\right)$
$-10$
$\begin{array}{c}\mathrm{lim}\\ {}^{x\to 4}\end{array}\frac{7-\sqrt{12x+1}}{x-4}$
$\underset{x\to -3}{\mathrm{lim}}\left(\frac{\frac{1}{3}+\frac{1}{x}}{3+x}\right)$
$-\frac{1}{9}$
For the following exercises, use numerical evidence to determine whether the limit exists at $\text{\hspace{0.17em}}x=a.\text{\hspace{0.17em}}$ If not, describe the behavior of the graph of the function at $\text{\hspace{0.17em}}x=a.$
$f(x)=\frac{-2}{x-4};\text{}a=4$
$f(x)=\frac{-2}{{\left(x-4\right)}^{2}};\text{}a=4$
At $\text{\hspace{0.17em}}x=4,$ the function has a vertical asymptote.
$f(x)=\frac{-x}{{x}^{2}-x-6};\text{}a=3$
$f(x)=\frac{6{x}^{2}+23x+20}{4{x}^{2}-25};\text{}a=-\frac{5}{2}$
removable discontinuity at $\text{\hspace{0.17em}}a=-\frac{5}{2}$
$f(x)=\frac{\sqrt{x}-3}{9-x};\text{}a=9$
For the following exercises, determine where the given function $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.
$f(x)={x}^{2}-2x-15$
continuous on $\text{\hspace{0.17em}}(-\infty ,\infty )$
$f(x)=\frac{{x}^{2}-2x-15}{x-5}$
$f(x)=\frac{{x}^{2}-2x}{{x}^{2}-4x+4}$
removable discontinuity at $\text{\hspace{0.17em}}x=2.$ $\text{\hspace{0.17em}}f(2)\text{\hspace{0.17em}}$ is not defined, but limits exist.
$f(x)=\frac{{x}^{3}-125}{2{x}^{2}-12x+10}$
$f(x)=\frac{{x}^{2}-\frac{1}{x}}{2-x}$
discontinuity at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$ Both $\text{\hspace{0.17em}}f(0)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(2)\text{\hspace{0.17em}}$ are not defined.
$f(x)=\frac{x+2}{{x}^{2}-3x-10}$
$f(x)=\frac{x+2}{{x}^{3}+8}$
removable discontinuity at $\text{\hspace{0.17em}}x=\u20132.\text{}f(\u20132)\text{\hspace{0.17em}}$ is not defined.
For the following exercises, find the average rate of change $\text{\hspace{0.17em}}\frac{f(x+h)-f(x)}{h}.$
$f(x)=3x+2$
$f(x)=\frac{1}{x+1}$
$f(x)=\mathrm{ln}(x)$
$\frac{\mathrm{ln}(x+h)-\mathrm{ln}(x)}{h}$
$f(x)={e}^{2x}$
For the following exercises, find the derivative of the function.
$f(x)=5{x}^{2}-3x$
Find the equation of the tangent line to the graph of
$\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ at the indicated
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ value.
$f(x)=-{x}^{3}+4x$ ;
$\text{\hspace{0.17em}}x=2.$
$y=-8x+16$
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.
$f(x)=\frac{x}{\left|x\right|}$
Given that the volume of a right circular cone is $\text{\hspace{0.17em}}V=\frac{1}{3}\pi {r}^{2}h\text{\hspace{0.17em}}$ and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of $\text{\hspace{0.17em}}\pi $
$12\pi $
For the following exercises, use the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] .
$\underset{x\to {\mathrm{-1}}^{+}}{\mathrm{lim}}f(x)$
$\underset{x\to \mathrm{-1}}{\mathrm{lim}}f(x)$
$\underset{x\to \mathrm{-2}}{\mathrm{lim}}f(x)$
$\mathrm{-1}$
At what values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ If the function has a limit as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a,$ state it. If not, discuss why there is no limit
$f(x)=\{\begin{array}{ll}\frac{1}{x}-3,\text{i}f\hfill & x\le 2\hfill \\ {x}^{3}+1,if\hfill & x2\hfill \end{array}\text{}a=2$
$\underset{x\to {2}^{-}}{\mathrm{lim}}f(x)=-\frac{5}{2}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\underset{x\to {2}^{+}}{\mathrm{lim}}f(x)=9\text{\hspace{0.17em}}$ Thus, the limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 2 does not exist.
$f(x)=\{\begin{array}{lll}{x}^{3}+1,\hfill & if\hfill & x<1\hfill \\ 3{x}^{2}-1,\hfill & if\hfill & x=1\hfill \\ -\sqrt{x+3}+4,\hfill & if\hfill & x>1\hfill \end{array}\text{}a=1$
For the following exercises, evaluate each limit using algebraic techniques.
$\underset{x\to \mathrm{-5}}{\mathrm{lim}}\left(\frac{\frac{1}{5}+\frac{1}{x}}{10+2x}\right)$
$-\frac{1}{50}$
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{\sqrt{{h}^{2}+25}-5}{{h}^{2}}\right)$
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{1}{h}-\frac{1}{{h}^{2}+h}\right)$
1
For the following exercises, determine whether or not the given function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.
$f(x)=\sqrt{{x}^{2}-4}$
$f(x)=\frac{{x}^{3}-4{x}^{2}-9x+36}{{x}^{3}-3{x}^{2}+2x-6}$
removable discontinuity at $\text{\hspace{0.17em}}x=3$
For the following exercises, use the definition of a derivative to find the derivative of the given function at $\text{\hspace{0.17em}}x=a.$
$f(x)=\frac{3}{5+2x}$
$f(x)=\frac{3}{\sqrt{x}}$
$f\text{'}(x)=-\frac{3}{2{a}^{\frac{3}{2}}}$
$f(x)=2{x}^{2}+9x$
For the graph in [link] , determine where the function is continuous/discontinuous and differentiable/not differentiable.
discontinuous at –2,0, not differentiable at –2,0, 2.
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.
$f(x)=\left|x-2\right|-\left|x+2\right|$
$f(x)=\frac{2}{1+{e}^{\frac{2}{x}}}$
not differentiable at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ (no limit)
For the following exercises, explain the notation in words when the height of a projectile in feet, $\text{\hspace{0.17em}}s,$ is a function of time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ in seconds after launch and is given by the function $\text{\hspace{0.17em}}s(t).$
$s(2)$
the height of the projectile at $\text{\hspace{0.17em}}t=2\text{\hspace{0.17em}}$ seconds
$s\text{'}(2)$
$\frac{s(2)-s(1)}{2-1}$
the average velocity from $\text{\hspace{0.17em}}t=1\text{to}t=2$
For the following exercises, use technology to evaluate the limit.
$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}(x)}{3x}$
$\frac{1}{3}$
$\underset{x\to 0}{\mathrm{lim}}\frac{{\mathrm{tan}}^{2}(x)}{2x}$
$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}(x)(1-\mathrm{cos}(x))}{2{x}^{2}}$
0
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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the function below discontinuous?
$f(x)=\{\begin{array}{c}4x-7\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ne 1\\ {x}^{2}-4\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=1\end{array}$
For the following exercises, consider the function whose graph appears in [link] .
Find the average rate of change of the function from $\text{\hspace{0.17em}}x=1\text{to}x=3.$
2
Find all values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at which $\text{\hspace{0.17em}}f\text{'}(x)=0.$
$x=1$
Find all values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at which $\text{\hspace{0.17em}}f\text{'}(x)\text{\hspace{0.17em}}$ does not exist.
Find an equation of the tangent line to the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ the indicated point: $\text{\hspace{0.17em}}f(x)=3{x}^{2}-2x-6,\text{}x=-2$
$y=-14x-18$
For the following exercises, use the function $\text{\hspace{0.17em}}f(x)=x{\left(1-x\right)}^{\frac{2}{5}}$ .
Graph the function $\text{\hspace{0.17em}}f(x)=x{\left(1-x\right)}^{\frac{2}{5}}\text{\hspace{0.17em}}$ by entering $\text{\hspace{0.17em}}f(x)=x{\left({\left(1-x\right)}^{2}\right)}^{\frac{1}{5}}\text{\hspace{0.17em}}$ and then by entering $\text{\hspace{0.17em}}f(x)=x{\left({\left(1-x\right)}^{\frac{1}{5}}\right)}^{2}$ .
Explore the behavior of the graph of $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ around $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=1.$
The graph is not differentiable at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ (cusp).
For the following exercises, find the derivative of each of the functions using the definition: $\text{\hspace{0.17em}}\underset{h\to 0}{\mathrm{lim}}\frac{f(x+h)-f(x)}{h}$
$f(x)=2x-8$
$f(x)=x-\frac{1}{2}{x}^{2}$
$f(x)=\frac{1}{x+2}$
${f}^{\text{'}}(x)=-\frac{1}{{\left(2+x\right)}^{2}}$
$f(x)=\frac{3}{x-1}$
$f(x)={x}^{2}+{x}^{3}$
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