# Virtual Math Learning Center

## Practice Problems for Module 3

### Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. Evaluate the limit.

1. $$\displaystyle \lim_{x\rightarrow -3}\frac{x^2-x-12}{x+3}$$
2. $$\displaystyle \lim_{x\rightarrow 1}\frac{x^2-x-2}{x+1}$$
3. $$\displaystyle \lim_{t\rightarrow 0}\frac{\sqrt{2-t}-\sqrt{2}}{t}$$
4. $$\displaystyle \lim_{h\rightarrow 0}\frac{(3+h)^{-1}-3^{-1}}{h}$$
5. $$\displaystyle \lim_{t\rightarrow 1}\left\langle 2t-3, \frac{t^2-t}{t-1}\right\rangle$$

1. $$-7$$
2. $$-1$$
3. $$-\dfrac{1}{2\sqrt{2}}$$
4. $$-\dfrac{1}{9}$$
5. $$\langle -1,1\rangle$$

2. Find the limit.

1. $$\displaystyle \lim_{x\rightarrow -4^-}\frac{|x+4|}{x+4}$$
2. $$\displaystyle \lim_{x\rightarrow -4^+}\frac{|x+4|}{x+4}$$
3. $$\displaystyle \lim_{x\rightarrow -4}\frac{|x+4|}{x+4}$$

1. $$-1$$
2. $$1$$
3. DNE

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3. For the function below, evaluate each of the following limits if it exists.
$f(x)=\left\{ \begin{array}{ll} x & \textrm{if } x<0\\ x^2 & \textrm{if } 0<x\leq 2\\ 8-x & \textrm{if } x>2\\ \end{array}\right.$

1. $$\displaystyle \lim_{x\rightarrow 0^+} f(x)$$
2. $$\displaystyle \lim_{x\rightarrow 0^-} f(x)$$
3. $$\displaystyle \lim_{x\rightarrow 0} f(x)$$
4. $$\displaystyle \lim_{x\rightarrow 1} f(x)$$
5. $$\displaystyle \lim_{x\rightarrow 2^-} f(x)$$
6. $$\displaystyle \lim_{x\rightarrow 2^+} f(x)$$
7. $$\displaystyle \lim_{x\rightarrow 2} f(x)$$

1. $$0$$
2. $$0$$
3. $$0$$
4. $$1$$
5. $$4$$
6. $$6$$
7. DNE

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4. Explain why the following function is discontinuous at $$x=0$$.
$f(x)=\left\{ \begin{array}{ll} \cos x & \textrm{if } x<0\\ 0 & \textrm{if } x=0\\ 1-x^2 & \textrm{if } x>0\\ \end{array}\right.$ ​

Since $$f(0)=0$$ and $$\displaystyle \lim_{x\rightarrow 0} f(x)=1$$, then $$f(x)$$ is NOT continuous at $$x=0$$ because $$f(0)\neq \displaystyle \lim_{x\rightarrow 0} f(x).$$

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5. Find the values of $$a$$ and $$b$$ that make $$f$$ continuous everywhere.
$f(x)=\left\{ \begin{array}{ll} \dfrac{x^2-4}{x-2} & \textrm{if } x<2\\ ax^2-bx+3 & \textrm{if } 2\leq x <3\\ 2x-a+b & \textrm{if } x\geq 3\\ \end{array}\right.$

$$a=\dfrac{1}{2}, \qquad b=\dfrac{1}{2}$$

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6. Find the limit.

1. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{1-x^2}{x^3-x+1}$$
2. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{4x^3+6x^2-2}{2x^3-4x+5}$$
3. $$\displaystyle \lim_{x\rightarrow \infty} \left( \sqrt{9x^2+x}-3x\right)$$
4. $$\displaystyle \lim_{x\rightarrow -\infty} \dfrac{1+x^6}{x^4+1}$$
5. $$\displaystyle \lim_{x\rightarrow- \infty} \dfrac{\sqrt{1+4x^6}}{2-x^3}$$
6. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{\sqrt{x+3x^2}}{4x-1}$$
7. $$\displaystyle \lim_{x\rightarrow \infty} \dfrac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$$
8. $$\displaystyle \lim_{x\rightarrow -\infty} \dfrac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$$

1. 0
2. 2
3. $$\dfrac{1}{6}$$
4. $$+\infty$$
5. 2
6. $$\dfrac{\sqrt{3}}{4}$$
7. 1
8. $$-1$$