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Math Help - show limit of e^(1/n) / n = 0

  1. #1
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    show limit of e^(1/n) / n = 0

    Hi all again,

    I know that this limit = 0, but I'm blanking on how to show it..

    \lim_{n\to\infty} \frac{e^{1/n}}{n} = 0

    I'm trying to manipulate it into an acceptable form for l'Hospital, but it isn't coming to me. Thanks for any help!

    Brian
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  2. #2
    MHF Contributor Calculus26's Avatar
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    don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0
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  3. #3
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    Quote Originally Posted by Calculus26 View Post
    don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0
    Oh! Of course...duh.. heh, thanks.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by buckeye1973 View Post
    Hi all again,

    I know that this limit = 0, but I'm blanking on how to show it..

    \lim_{n\to\infty} \frac{e^{1/n}}{n} = 0

    I'm trying to manipulate it into an acceptable form for l'Hospital, but it isn't coming to me. Thanks for any help!

    Brian
    Quote Originally Posted by Calculus26 View Post
    don't need L'Hopital as directly you get e^(0)/inf = 1/inf = 0
    A little more rigorously note that \left|\frac{e^{\frac{1}{n}}}{n}\right|\leqslant\fr  ac{1}{n}
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    A little more rigorously note that \left|\frac{e^{\frac{1}{n}}}{n}\right|\leqslant\fr  ac{1}{n}

    So we will apply the sandwich theorem, right ?
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by TWiX View Post
    So we will apply the sandwich theorem, right ?
    True dat.
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