 # Practice Problems for Module 3

Sections 2.3, 2.5, and 2.6

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
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. $$-1$$