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Mathematics LibreTexts

1.1 E Exercises

  • Page ID
    10658
  • [ "stage:draft", "article:topic" ]

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    Exercise \(\PageIndex{1}\) Terms and Concepts

    1.  What are the  ways in which a limit may fail to exist?
    2.  T/F: If \(\lim\limits_{x\to1-}f(x)=5\), then \(\lim\limits_{x\to1}f(x)=5\)
    3.  T/F: If \(\lim\limits_{x\to1-}f(x)=5\), then \(\lim\limits_{x\to1+}f(x)=5\)
    4.  T/F: If \(\lim\limits_{x\to1}f(x)=5\), then \(\lim\limits_{x\to1-}f(x)=5\)
    5. T/F: If \(\lim\limits_{x\to 1^-}f(x)=-\infty\), then \(\lim\limits_{x\to 1^+}f(x)=\infty\).
    6.  T/F: If \(\lim\limits_{x\to 5}f(x)=\infty\), then \(f\) has a vertical asymptote at \(x=5\).

     

    Answer

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    Exercise \(\PageIndex{2}\): Finding limits using Graphs

    Evaluate each expression using the given graph of \(f(x)\).


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    Answer

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    Exercise \(\PageIndex{3}\): Finding limits using Graphs

    Evaluate each expression using the given graph of \(f(x)\).

    146.PNG

    Answer

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    Exercise \(\PageIndex{4}\): Finding limits using Graphs

    147.PNG

     

     

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    Exercise \(\PageIndex{5}\): Finding limits using Graphs

    148.PNG

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    Exercise \(\PageIndex{6}\): Finding limits using Graphs

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    Exercise \(\PageIndex{7}\): Finding limits using Graphs

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    Exercise \(\PageIndex{8}\): Finding limits using Graphs

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    Exercise \(\PageIndex{9}\): Infinite limits

    In Exercises 1-6, evaluate the given limits using the graph of the function.

    1. \(f(x) = \frac{1}{(x+1)^2}\)
    (a) \(\lim\limits_{x\to -1^-}f(x)\)
    (b) \(\lim\limits_{x\to -1^+}f(x)\)
    169.PNG

    2. \(f(x) = \frac{1}{(x-3)(x-5)^2}\)
    (a) \(\lim\limits_{x\to 3^-}f(x)\)
    (b) \(\lim\limits_{x\to 3^+}f(x)\)
    (c) \(\lim\limits_{x\to 3}f(x)\)
    (d) \(\lim\limits_{x\to 5^-}f(x)\)
    (e) \(\lim\limits_{x\to 5^+}f(x)\)
    (f) \(\lim\limits_{x\to 5}f(x)\)
    1610.PNG

    3. \(f(x) = \frac{1}{e^x+1}\)
    (a) \(\lim\limits_{x\to -\infty}f(x)\)
    (b) \(\lim\limits_{x\to \infty}f(x)\)
    (c) \(\lim\limits_{x\to 0^-}f(x)\)
    (d) \(\lim\limits_{x\to 0^+}f(x)\)
    1611.PNG

    4. \(f(x) = x^2\sin (\pi x)\)
    (a) \(\lim\limits_{x\to -\infty}f(x)\)
    (b) \(\lim\limits_{x\to \infty}f(x)\)
    1612.PNG

    5. \(f(x)=\cos (x)\)
    (a) \(\lim\limits_{x\to -\infty}f(x)\)
    (b) \(\lim\limits_{x\to \infty}f(x)\)
    1613.PNG

    6. \(f(x) = 2^x +10\)
    (a) \(\lim\limits_{x\to -\infty}f(x)\)
    (b) \(\lim\limits_{x\to \infty}f(x)\)
    1614.PNG

    Answer

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    Exercise \(\PageIndex{10}\): One sided limits

    Given \( \displaystyle f(x)= \left\{ \begin{array}{ccc}
    x^3+1 & \mbox{ if } x <0 \\
    0 & \mbox{ if } x =0 \\
    \sqrt{x+1}-2 & \mbox{ if } x >0 \\
    \end{array}
    \right.\)

     Identify each limit:

    1. \( \displaystyle \lim_{x \to 0^-} f(x)\)
    2. \( \displaystyle \lim_{x \to 0^+} f(x)\)
    3. \( \displaystyle \lim_{x \to 0} f(x)\)
    4. \( \displaystyle \lim_{x \to -1} f(x)\)
    5. \( \displaystyle \lim_{x \to 3} f(x)\)
    Hint:

    Draw a graph.

    Answer:

    11, -1, DNE, 0, 0

    Solution:
    1. \( \displaystyle \lim_{x \to 0^-} f(x)=0^3+11\)
    2. \( \displaystyle \lim_{x \to 0^+} f(x)= \sqrt{0+1}-2=-1 \)
    3. Since \( \displaystyle \lim_{x \to 0^-} f(x) \ne \displaystyle \lim_{x \to 0^+} f(x)\), \( \displaystyle \lim_{x \to 0} f(x) =\)  DNE. \
    4. \(\displaystyle \lim_{x \to -1} f(x)=(-1)^3+1=0\)
    5. \( \displaystyle \lim_{x \to 3} f(x)= \sqrt{3+1}-2=0\)

    Exercise \(\PageIndex{11}\): Infinite limits

    \(\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}\)

    Answer

    Since \(\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36}= \frac{12}{0}\), and \( \frac{5.9+6}{(5.9)^2-36}<0,\)

    \(\displaystyle\lim_{y \to 6^-} \frac{y+6}{y^2-36} =-\infty\).

    Contributors

    Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/