Limit of a sequence

dobry_den

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!

NateTG

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.

dobry_den

The subsequence way of solving is elegant, i didn't realise it... thanks a lot!