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Math Help - Infinite Limit involving Natural Log w/ Exponent

  1. #1
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    Infinite Limit involving Natural Log w/ Exponent

    Alright, I'm not the greatest mathematician so I thought I would ask for help on this one, it seems a bit more complicated than I'm used to.

    as limit x approaches +infinity (ln x)^(1/x)

    Any thoughts?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by gnelly08 View Post
    Alright, I'm not the greatest mathematician so I thought I would ask for help on this one, it seems a bit more complicated than I'm used to.

    as limit x approaches +infinity (ln x)^(1/x)

    Any thoughts?
    It's 1, consider the limit of ln(ln(x)^(1/x))=(1/x)ln(ln(x)).

    CB
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  3. #3
    Junior Member Infophile's Avatar
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    Hello,

    There are several methods, I suggest you this :

    \ln(x)^{\frac{1}{x}}=\exp\left(\frac{\ln(\ln(x))}{  x}\right)

    We want to find the limit of \frac{\ln(\ln(x))}{x} when x\to \infty.

    Take y=\ln(x) then \frac{\ln(y)}{e^y}.

    And for y\to \infty we have 0\le \frac{\ln(y)}{e^y}\le \frac{y-1}{e^y}\longrightarrow 0

    So \lim_{y\to \infty} \frac{\ln(y)}{e^y}=0 and therefore \color{red}\boxed{\lim_{x\to +\infty}\ln(x)^{\frac{1}{x}}=1}

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  4. #4
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    Alright, I follow you both. =) Thanks for the help.
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