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Math Help - Help with evaluating limits

  1. #1
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    Help with evaluating limits

    Hi!

    I need to find the limit of the following:

    \lim_{x\rightarrow\infty}\frac{x^k}{e^x} while k\in\mathbb{N}.

    I know it's 0 since exponential functions grows faster then any polynomial functions, but I couldn't find a way to prove it.

     \lim_{x\rightarrow\infty}\frac{x^\ln x}{\ln ^xx}

    Please don't post a solution, I just need guidance and hints on what should I try.

    Thanks in advanced!
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  2. #2
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    Re: Help with evaluating limits

    The first one can be proven using L'Hospital's Rule "k" times.
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  3. #3
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    Re: Help with evaluating limits

    Hi Prove It!
    Thanks for the help.

    That's brilliant.
    didn't think about it.
    \lim_{x\rightarrow \infty}\frac{x^k}{e^x}=\lim_{x\rightarrow \infty}\frac{kx^{k-1}}{e^x}=\lim_{x\rightarrow \infty}\frac{k(k-1)x^{k-2}}{e^x}=...=\lim_{x\rightarrow \infty}\frac{k!}{e^x}=0
    (correct me if there's an error)

    what about the second one?
    any thoughts?
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