Evaluate $\lim \limits_{x \to \infty}\frac{\ln(x)}{x^a}$

Asked Dec 29 '21 at 10:02

Active Dec 29 '21 at 10:14

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I should calculate this expression, with $a > 0$:
$$\lim \limits_{x \to \infty}\frac{\ln(x)}{x^a}.$$

But I am stuck at this equation for quite some time. Its a indeterminate form but we are not yet allowed to use the L'Hôpital's rule. We definded $x^a = e^{a\ln(a)}$.

I tried some things such as sandwich theorem, but it didnt work. Overall I just have no idea how to solve this.
Help would be appreciated.

edited Dec 29 '21 at 10:09

Gary

31,845

asked Dec 29 '21 at 10:02

Lupos

What do think the limit is? And why? This is always a good starting point for computing a limit. – Jacobiman Dec 29 '21 at 10:07
This has been asked a million times before on this site. – Gary Dec 29 '21 at 10:09
why is he in the second answer allowed to replace x^a with e^y? – Lupos Dec 29 '21 at 10:57

Evaluate $\lim \limits_{x \to \infty}\frac{\ln(x)}{x^a}$

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