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Math Help - limit question

  1. #1
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    limit question

    what is the limit of (ln^2 k)^{\frac{1}{k}}
    when k goes to infinity
    ?
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  2. #2
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    Re: limit question

    Quote Originally Posted by transgalactic View Post
    what is the limit of (ln^2 k)^{\frac{1}{k}}
    when k goes to infinity
    ?
    \displaystyle \begin{align*}\lim_{k \to \infty}\left(\ln^2{k}\right)^{\frac{1}{k}} &= \lim_{k \to \infty}e^{\ln{\left[\left(\ln^2{k}\right)^{\frac{1}{k}}\right]}} \\ &= e^{\lim_{k \to \infty}\ln{\left[\left(\ln^2{k}\right)^{\frac{1}{k}}\right]}} \\ &= e^{\lim_{k \to \infty}\left[\frac{\ln{\left(\ln^2{k}\right)}}{k}\right]}\end{align*}

    Now you can apply L'Hospital's Rule.
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  3. #3
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    Re: limit question

    Quote Originally Posted by transgalactic View Post
    what is the limit of (ln^2 k)^{\frac{1}{k}} when k goes to infinity?
    This can also be done by direct comparison.

    If k\ge 3 then we get \ln(k)\le 2\sqrt{k}.

    Thus 1 \leqslant \sqrt[k]{{\ln ^2 (k)}} \leqslant \sqrt[k]{{4k}}
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  4. #4
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    Re: limit question

    Quote Originally Posted by transgalactic View Post
    what is the limit of (ln^2 k)^{\frac{1}{k}}
    when k goes to infinity
    ?
    Could you please clarify your notation, does \log^2k denote (\log(k))^2 or \log(\log(k)) ?

    Not that it makes a jot of difference to the answer!

    CB
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    Re: limit question

    Quote Originally Posted by CaptainBlack View Post
    Could you please clarify your notation, does \log^2k denote (\log(k))^2 or \log(\log(k)) ?

    Not that it makes a jot of difference to the answer!

    CB
    I expect it's being used in the same context as \displaystyle \sin^2{x} for \displaystyle (\sin{x})^2...
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  6. #6
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    Re: limit question

    Quote Originally Posted by Prove It View Post
    I expect it's being used in the same context as \displaystyle \sin^2{x} for \displaystyle (\sin{x})^2...
    So do I, but I object to the abuse of notation in this case because the iterated logarithm is a relativly common function.

    (as it happens I also object to the systematic omission of parentheses around the arguments of functions in elementary mathematics)

    CB
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