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Mathematics LibreTexts

1.3E Exercises

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    10659
  • [ "stage:draft", "article:topic" ]

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    Exercise \(\PageIndex{1}\) 

    Using:

    \[\begin{align}\lim\limits_{x\to9}f(x)=6 \qquad \lim\limits_{x\to6}f(x)=9 \\ \lim\limits_{x\to9}g(x)=3 \qquad \lim\limits_{x\to6}g(x)=3 \end{align}\]

    evaluate the limits given in Exercises 6-13, where possible. If it is not possible to know, state so.

    1.  \(\lim\limits_{x\to9}(f(x)+g(x))\)
    2.  \(\lim\limits_{x\to9}(3f(x)/g(x))\)
    3.  \(\lim\limits_{x\to9} \left ( \frac{f(x)-2g(x)}{g(x)}\right )\)
    4. \(\lim\limits_{x\to6}\left (\frac{f(x)}{3-g(x)}\right )\)
    5. \(\lim\limits_{x\to9}g(f(x))\) \(\lim\limits_{x\to6}f(g(x))\)
    6.  \(\lim\limits_{x\to6}g(f(f(x)))\)
    7.  \(\lim\limits_{x\to6}f(x)g(x)-f^2(x)+g^2(x)\)

    Using

    \[\begin{align}\lim\limits_{x\to1}f(x)=2 \qquad \lim\limits_{x\to10}f(x)=1 \\ \lim\limits_{x\to1}g(x)=0 \qquad \lim\limits_{x\to10}g(x)=\pi \end{align}\]

    evaluate the limits given in Exercises 14-17, where possible. If it is not possible to know, state so.

    9. \(\lim\limits_{x\to1}f(x)^{g(x)}\)

    10. \(\lim\limits_{x\to10}\cos (g(x))\)

    11. \(\lim\limits_{x\to1}f(x)g(x)\)

    12. \(\lim\limits_{x\to1}g(5f(x))\)

    Answer

    Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page.

    Exercise \(\PageIndex{2}\)

    Evaluate the following:

    1. \(\lim\limits_{x\to3}x^2-3x+7\)
    2.  \(\lim\limits_{x\to\pi}\left ( \frac{x-3}{x+5}\right )^7\)
    3. \(\lim\limits_{x\to\pi /4}\cos x \sin x\)
    4.  \(\lim\limits_{x\to 0}\ln x\)
    5.  \(\lim\limits_{x\to3}4^{{x^3}-8x}\)
    6.  \(\lim\limits_{x\to\pi/6}\csc x\)
    7.  \(\lim\limits_{x\to0}\ln (1+x)\)
    8.  \(\lim\limits_{x\to\pi}\frac{x^2+3x+5}{5x^2-2x-3}\)
    9. \(\lim\limits_{x\to\pi}\frac{3x+1}{1-x}\)
    10. \(\lim\limits_{x\to6}\frac{x^2-4x-12}{x^2-13x+42}\)
    11. \(\lim\limits_{x\to0}\frac{x^2+2x}{x^2-2x}\)
    12. \(\lim\limits_{x\to2}\frac{x^2+6x-16}{x^2-3x+2}\)
    13. \(\lim\limits_{x\to2}\frac{x^2-5x-14}{x^2+10x+16}\)
    14. \(\lim\limits_{x\to-2}\frac{x^2-5x-14}{x^2+10x+16}\)
    15. \(\lim\limits_{x\to-1}\frac{x^2+9x+8}{x^2-6x-7}\)
    Answer

    Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page.

     

    Exercise \(\PageIndex{3}\): Limit with indeterminate form

    \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x+4}-2}{x}\)

    Answer

    \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x+4}-2}{x} = \frac{\sqrt{0+4}-2}{0} =\left[\frac{0}{0}\right]\)

                                                                                   = \( \displaystyle \lim_{x \to 0} \frac{(\sqrt{x+4}-2) (\sqrt{x+4}+2)}{x (\sqrt{x+4}+2)}\)

                                                                                   = \( \displaystyle \lim_{x \to 0} \frac{((x+4)-4) }{x (\sqrt{x+4}+2)}\)

                                                                                    = \( \displaystyle \lim_{x \to 0} \frac{x }{x (\sqrt{x+4}+2)}\)

                                                                                     = \( \displaystyle \lim_{x \to 0} \frac{1 }{(\sqrt{x+4}+2)}= \frac{1 }{(\sqrt{0+4}+2)}= \frac{1 }{4}\).

    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/

    Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)