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Access the following online resource for additional instruction and practice with properties of limits.
Give an example of a type of function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ whose limit, as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a,$ is $\text{\hspace{0.17em}}f\left(a\right).$
If $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a polynomial function, the limit of a polynomial function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ will always be $\text{\hspace{0.17em}}f\left(a\right).$
When direct substitution is used to evaluate the limit of a rational function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and the result is $\text{\hspace{0.17em}}f\left(a\right)=\frac{0}{0},$ does this mean that the limit of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ does not exist?
What does it mean to say the limit of $\text{\hspace{0.17em}}f\left(x\right),$ as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}c,$ is undefined?
It could mean either (1) the values of the function increase or decrease without bound as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}c,$ or (2) the left and right-hand limits are not equal.
For the following exercises, evaluate the limits algebraically.
$\underset{x\to 0}{\mathrm{lim}}\left(3\right)$
$\underset{x\to 2}{\mathrm{lim}}\left(\frac{-5x}{{x}^{2}-1}\right)$
$\frac{-10}{3}$
$\underset{x\to 2}{\mathrm{lim}}\left(\frac{{x}^{2}-5x+6}{x+2}\right)$
$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}-9}{x-3}\right)$
6
$\underset{x\to -1}{\mathrm{lim}}\left(\frac{{x}^{2}-2x-3}{x+1}\right)$
$\underset{x\to \frac{3}{2}}{\mathrm{lim}}\left(\frac{6{x}^{2}-17x+12}{2x-3}\right)$
$\frac{1}{2}$
$\underset{x\to -\frac{7}{2}}{\mathrm{lim}}\left(\frac{8{x}^{2}+18x-35}{2x+7}\right)$
$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}-9}{x-5x+6}\right)$
6
$\underset{x\to -3}{\mathrm{lim}}\left(\frac{-7{x}^{4}-21{x}^{3}}{-12{x}^{4}+108{x}^{2}}\right)$
$\underset{x\to 3}{\mathrm{lim}}\left(\frac{{x}^{2}+2x-3}{x-3}\right)$
does not exist
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(3+h\right)}^{3}-27}{h}\right)$
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(2-h\right)}^{3}-8}{h}\right)$
$-12$
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(h+3\right)}^{2}-9}{h}\right)$
$\underset{h\to 0}{\mathrm{lim}}\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right)$
$-\frac{\sqrt{5}}{10}$
$\underset{x\to 0}{\mathrm{lim}}\left(\frac{\sqrt{3-x}-\sqrt{3}}{x}\right)$
$\underset{x\to 9}{\mathrm{lim}}\left(\frac{{x}^{2}-81}{3-\sqrt{x}}\right)$
$-108$
$\underset{x\to 1}{\mathrm{lim}}\left(\frac{\sqrt{x}-{x}^{2}}{1-\sqrt{x}}\right)$
$\underset{x\to 0}{\mathrm{lim}}\left(\frac{x}{\sqrt{1+2x}-1}\right)$
1
$\underset{x\to \frac{1}{2}}{\mathrm{lim}}\left(\frac{{x}^{2}-\frac{1}{4}}{2x-1}\right)$
$\underset{x\to 4}{\mathrm{lim}}\left(\frac{{x}^{3}-64}{{x}^{2}-16}\right)$
6
$\underset{x\to {2}^{-}}{\mathrm{lim}}\left(\frac{|x-2|}{x-2}\right)$
$\underset{x\to {2}^{+}}{\mathrm{lim}}\left(\frac{\left|x-2\right|}{x-2}\right)$
1
$\underset{x\to 2}{\mathrm{lim}}\left(\frac{\left|x-2\right|}{x-2}\right)$
$\underset{x\to {4}^{-}}{\mathrm{lim}}\left(\frac{\left|x-4\right|}{4-x}\right)$
1
$\underset{x\to {4}^{+}}{\mathrm{lim}}\left(\frac{\left|x-4\right|}{4-x}\right)$
$\underset{x\to 4}{\mathrm{lim}}\left(\frac{\left|x-4\right|}{4-x}\right)$
does not exist
$\underset{x\to 2}{\mathrm{lim}}\left(\frac{-8+6x-{x}^{2}}{x-2}\right)$
For the following exercise, use the given information to evaluate the limits: $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}f(x)=3,$ $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}g\left(x\right)=5$
$\underset{x\to c}{\mathrm{lim}}\text{\hspace{0.17em}}\left[\text{\hspace{0.17em}}2f(x)+\sqrt{g(x)}\text{\hspace{0.17em}}\right]$
$6+\sqrt{5}$
$\underset{x\to c}{\mathrm{lim}}\text{\hspace{0.17em}}\left[\text{\hspace{0.17em}}3f(x)+\sqrt{g(x)}\text{\hspace{0.17em}}\right]$
$\underset{x\to c}{\mathrm{lim}}\frac{f(x)}{g(x)}$
$\frac{3}{5}$
For the following exercises, evaluate the following limits.
$\underset{x\to 2}{\mathrm{lim}}\mathrm{cos}\left(\pi x\right)$
$\underset{x\to 2}{\mathrm{lim}}\mathrm{sin}\left(\pi x\right)$
0
$\underset{x\to 2}{\mathrm{lim}}\mathrm{sin}\left(\frac{\pi}{x}\right)$
$f(x)=\{\begin{array}{ll}2{x}^{2}+2x+1,\hfill & x\le 0\hfill \\ x-3,\hfill & x0\hfill \end{array}\text{;}\underset{x\to {0}^{+}}{\mathrm{lim}}f(x)$
$-3$
$f(x)=\{\begin{array}{ll}2{x}^{2}+2x+1,\hfill & x\le 0\hfill \\ x-3,\hfill & x0\hfill \end{array}\text{;}\underset{x\to {0}^{-}}{\mathrm{lim}}f(x)$
$f(x)=\{\begin{array}{ll}2{x}^{2}+2x+1,\hfill & x\le 0\hfill \\ x-3,\hfill & x0\hfill \end{array}\text{;}\underset{x\to 0}{\mathrm{lim}}f(x)$
does not exist; right-hand limit is not the same as the left-hand limit.
$\underset{x\to 4}{\mathrm{lim}}\frac{\sqrt{x+5}-3}{x-4}$
$\underset{x\to {2}^{+}}{\mathrm{lim}}(2x-\mathrm{\u301ax\u301b})$
2
$\underset{x\to 2}{\mathrm{lim}}\frac{\sqrt{x+7}-3}{{x}^{2}-x-2}$
$\underset{x\to {3}^{+}}{\mathrm{lim}}\frac{{x}^{2}}{{x}^{2}-9}$
Limit does not exist; limit approaches infinity.
For the following exercises, find the average rate of change $\text{\hspace{0.17em}}\frac{f(x+h)-f(x)}{h}.$
$f(x)=x+1$
$f(x)={x}^{2}+3x+4$
$f(x)=3{x}^{2}+1$
$f(x)=\mathrm{cos}(x)$
$\frac{\mathrm{cos}(x+h)-\mathrm{cos}(x)}{h}$
$f(x)=2{x}^{3}-4x$
$f(x)=\frac{1}{{x}^{2}}$
Find an equation that could be represented by [link] .
Find an equation that could be represented by [link] .
$f\left(x\right)=\frac{{x}^{2}+5x+6}{x+3}$
For the following exercises, refer to [link] .
What is the right-hand limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 0?
What is the left-hand limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 0?
does not exist
The position function $\text{\hspace{0.17em}}s(t)=-16{t}^{2}+144t\text{\hspace{0.17em}}$ gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval $\text{\hspace{0.17em}}\left[1,2\right]$ .
The height of a projectile is given by $\text{\hspace{0.17em}}s(t)=-64{t}^{2}+192t\text{\hspace{0.17em}}$ Find the average rate of change of the height from $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ second to $\text{\hspace{0.17em}}t=1.5\text{\hspace{0.17em}}$ seconds.
52
The amount of money in an account after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years compounded continuously at 4.25% interest is given by the formula $\text{\hspace{0.17em}}A={A}_{0}{e}^{0.0425t},$ where $\text{\hspace{0.17em}}{A}_{0}\text{\hspace{0.17em}}$ is the initial amount invested. Find the average rate of change of the balance of the account from $\text{\hspace{0.17em}}t=1\text{\hspace{0.17em}}$ year to $\text{\hspace{0.17em}}t=2\text{\hspace{0.17em}}$ years if the initial amount invested is $1,000.00.
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