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Math Help - What's the limit as x-> infinity?

  1. #1
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    What's the limit as x-> infinity?

    Prove that x^{1/\ln\ln{x}} \rightarrow \infty as x \rightarrow \infty.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Boysilver View Post
    Prove that x^{1/\ln\ln{x}} \rightarrow \infty as x \rightarrow \infty.

    Using the definition of limit ?.


    Fernando Revilla
    Last edited by FernandoRevilla; January 9th 2011 at 01:15 AM.
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    I'm guessing that you want to show this using L'Hopital's rule?

    Let y=x^{\frac{1}{\ln \ln x}}

    Take the \ln of both sides.

    \ln y=\frac{1}{\ln \ln x}\ln x.

    Now see if you can finish it with L'Hopital's Rule.
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by Boysilver View Post
    Prove that x^{1/\ln\ln{x}} \rightarrow \infty as x \rightarrow \infty.
    I have a 'handwaving' solution.

    Let x=10^{100}

    ln x=230 (approx.)

    ln ln x=ln 230=5.44 (approx.)

    \frac{1}{ln ln x}=\frac{1}{5.44}=0.18 (approx.)

    x^{\frac{1}{ln ln x}}=(10^{100})^{0.18}=10^{18}

    Therefore, x^{1/\ln\ln{x}} \rightarrow \infty as x \rightarrow \infty

    PS: This may not fetch you any marks in an exam but it will help you to 'think in limits'.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Boysilver View Post
    Prove that x^{1/\ln\ln{x}} \rightarrow \infty as x \rightarrow \infty.
    So, we are trying to ascertain \displaystyle \lim_{x\to\infty} x^{\frac{1}{\log(\log(x))}. Let \log(\log(x))=y so that our limit becomes \displaystyle \lim_{y\to\infty}(e^{e^{y})^{\frac{1}{y}} and thus to assume our limit existed would be to assume that \displaystyle \lim_{y\to\infty}\frac{e^y}{y} existed. More formally, x^{\frac{1}{\log(\log(x))}}=e^{\frac{\log(x)}{\log  (\log(x))}}\succ e^{\sqrt{\log(x)}}\to \infty where \succ means "asymptotically dominates"
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