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Limit of a sequence

  1. Jan 22, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the following limit:

    [tex]\lim_{n \rightarrow \infty}\frac{n}{\log_{10}{n}}[/tex]


    3. The attempt at a solution

    It's easy to find the limit using L'Hospital rule (after having used Heine theorem to transform the sequence into a function):

    [tex]\lim_{x \rightarrow \infty}\frac{x}{\log_{10}{x}} = \lim_{x \rightarrow \infty}\frac{1}{\frac{1}{x\log{10}}} = +\infty[/tex]

    Is there any way of solving it without L'Hospital rule?

    If I was to use the definition, then for every K, there should be such n_0 that for every n>n_0, (n/log_10(n)) > K. But I don't know how to solve this inequality. Any help would be greatly appreciated, thanks in advance!
     
    Last edited: Jan 22, 2008
  2. jcsd
  3. Jan 22, 2008 #2

    NateTG

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    Science Advisor
    Homework Helper

    There are easily manageable subsequences like:
    [tex]\frac{10^{10^k}}{\log_{10}10^{10^k}}=10^{10^k-k}[/tex]
    which clearly go to infinitely.

    You could also work through an epsilon-delta proof of l'Hospital's rule.
     
  4. Jan 22, 2008 #3
    The subsequence way of solving is elegant, i didn't realise it... thanks a lot!
     
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