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Math Help - help finding a limit

  1. #1
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    help finding a limit

    Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
    Thank you
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  2. #2
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    Quote Originally Posted by cowboys111 View Post
    Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
    Thank you
    I'm not sure how to prove it aside from just plugging in really large numbers, but \lim_{k \to \infty} k^{1/k} = 1.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by cowboys111 View Post
    Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
    Thank you
    L=\lim_{k\to\infty}\frac{\sqrt[k]{k}}{5}

    Taking the natural log of both sides gives

    \ln(L)=\lim_{k\to\infty}\frac{\ln(k)}{k}-\ln(5)

    Since this yields an indeterminate form we apply L'hopitals rule go get

    \ln(L)=\lim_{k\to\infty}\frac{\frac{1}{k}}{1}-\ln(5)=-\ln(5)

    So L=e^{-\ln(5)}=\frac{1}{5}
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by icemanfan View Post
    I'm not sure how to prove it aside from just plugging in really large numbers, but \lim_{k \to \infty} k^{1/k} = 1.
    Try considering

    \forall{x}>1,1\leq{x^{\frac{1}{x}}}\leq\bigg(1+\fr  ac{1}{\sqrt{x}}\bigg)
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  5. #5
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    thank you
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