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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

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #08


  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

      To see the full video page and find related videos, click the following link.
      MLC WIR 20B M151 week2 #09


  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 \(f(0)\neq \displaystyle \lim_{x\rightarrow 0} f(x)\).

    To see the full video page and find related videos, click the following link.
    MLC WIR 20B M151 week2 #10


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

    \(a=\dfrac{1}{2}, \qquad b=\dfrac{1}{2}\)

    To see the full video page and find related videos, click the following link.
    MLC WIR 20B M151 week2 #11


  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. \(-1\)