Limit of Sequence: Find w/o L'Hosp. Rule

[dobry_den]

Homework Statement

Find the following limit:

$$\lim_{n \rightarrow \infty}\frac{n}{\log_{10}{n}}$$

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):

$$\lim_{x \rightarrow \infty}\frac{x}{\log_{10}{x}} = \lim_{x \rightarrow \infty}\frac{1}{\frac{1}{x\log{10}}} = +\infty$$

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:
$$\frac{10^{10^k}}{\log_{10}10^{10^k}}=10^{10^k-k}$$
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 realize it... thanks a lot!