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Mathematics LibreTexts

2.9E Exercises

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  • Page ID
    10859
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    Exercise \(\PageIndex{1}\)

    For the following exercises, evaluate the limit.

    1) Evaluate the limit \(lim_{x→∞}\frac{e^x}{x}\).

    2) Evaluate the limit \(lim_{x→∞}\frac{e^x}{x^k}\).

    Answer

    \(∞\)

    3) Evaluate the limit \(lim_{x→∞}\frac{lnx}{x^k}\).

    4) Evaluate the limit \(lim_{x→a}\frac{x−a}{x^2−a^2}\).

    Answer

    \(\frac{1}{2a}\)

    5) Evaluate the limit \(lim_{x→a}\frac{x−a}{x^3−a^3}\).

    6) Evaluate the limit \(lim_{x→a}\frac{x−a}{x^n−a^n}\).

    Answer

    \(\frac{1}{na^{n−1}}\)

     

    Exercise \(\PageIndex{2}\)

    For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

    1) \(lim_{x→0^+}x^2lnx\)

    2) \(lim_{x→∞}x^{1/x}\)

    Answer

    Cannot apply directly; use logarithms

    3) \(lim_{x→0}x^{2/x}\)

    4) \(lim_{x→0}\frac{x^2}{1/x}\)

    Answer

    Cannot apply directly; rewrite as \(lim_{x→0}x^3\)

    5) \(lim_{x→∞}\frac{e^x}{x}\

     

    Exercise \(\PageIndex{3}\)

    For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.

    1) \(lim_{x→3}\frac{x^2−9}{x−3}\)

    Answer

    \(6\)

    2) \(lim_{x→3}\frac{x^2−9}{x+3}\)

    3) \(lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)

    Answer

    \(−2\)

    4) \(lim_{x→π/2}\frac{cosx}{\frac{π}{2}−x}\)

    5) \(lim_{x→π}\frac{x−π}{sinx}\)

    Answer

    \(−1\)

    6) \(lim_{x→1}\frac{x−1}{sinx}\)

    7) \(lim_{x→0}\frac{(1+x)^n−1}{x}\)

    Answer

    \(n\)

    8) \(lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)

    9) \(lim_{x→0}\frac{sinx−tanx}{x^3}\)

    Answer

    \(−\frac{1}{2}\)

    10) \(lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)

    11) \(lim_{x→0}\frac{e^x−x−1}{x^2}\)

    Answer

    \(\frac{1}{2}\)

    12) \(lim_{x→0}\frac{tanx}{\sqrt{x}}\)

    13) \(lim_{x→1}\frac{x→1}{lnx}\)

    Answer

    \(1\)

    14) \(lim_{x→0}(x+1)^{1/x}\)

    15) \(lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)

    Answer

    \(\frac{1}{6}\)

    16) \(lim_{x→0^+}x^{2x}\)

    17) \(lim_{x→∞}xsin(\frac{1}{x})\)

    Answer

    \(1\)

    18) \(lim_{x→0}\frac{sinx−x}{x^2}\)

    19) \(lim_{x→0^+}xln(x^4)\)

    Answer

    \(0\)

    20) \(lim_{x→∞}(x−e^x)\)

    21) \(lim_{x→∞}x^2e^{−x}\)

    Answer

    \(0\)

    22) \(lim_{x→0}\frac{3^x−2^x}{x}\)

    23) \(lim_{x→0}\frac{1+1/x}{1−1/x}\)

    Answer

    \(−1\)

    24) \(lim_{x→π/4}(1−tanx)cotx\)

    25) \(lim_{x→∞}xe^{1/}\)x

    Answer

    \(∞\)

    26) \(lim_{x→0}x^{1/cosx}\)

    27) \(lim_{x→0}x^{1/x}\)

    Answer

    \(1\)

    28) \(lim_{x→0}(1−\frac{1}{x})^x\)

    29) \(lim_{x→∞}(1−\frac{1}{x})^x\)

    Answer

    \(\frac{1}{e}\)

     

    Exercise \(\PageIndex{4}\)

    For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.

    1) \(lim_{x→0}\frac{e^x−1}{x}\)

    2) \(lim_{x→0}xsin(\frac{1}{x})\)

    Answer

    \(0\)

    3) \(lim_{x→1}\frac{x−1}{1−cos(πx)}\)

    4) \(lim_{x→1}\frac{e^{(x−1)}−1}{x−1}\)

    Answer

    \(1\)

    5) \(lim_{x→1}\frac{(x−1)^2}{lnx}\)

    6) \(lim_{x→π}\frac{1+cosx}{sinx}\)

    Answer

    \(0\)

    7) \(lim_{x→0}(cscx−\frac{1}{x})\)

    8) \(lim_{x→0^+}tan(x^x)\)

    Answer

    \(tan(1)\)

    9) \(lim_{x→0^+}\frac{lnx}{sinx}\)

    10) \(lim_{x→0}\frac{e^x−e^{−x}}{x}\)

    Answer

    \(2\)