Growth rate of logarithmic function?

Question

Just curious about the growth rate of the logarithmic function:

Does there exist a real number $n$ such that $lim_{x \to \infty} \frac{(ln(x))^{n}}{x}$ diverges (does not converge to $0$)?

Thanks in advance!

Answer 1

Answer is an emphatic no.

To see this, let $y=\ln(x)$. Then, as $x \rightarrow \infty$, $y \rightarrow \infty$,

and $\dfrac{\ln(x)^n}{x}=\dfrac{y^n}{e^y}$ which goes to $0$ as $y$ goes to $\infty$. (This can be seen easily using the L'Hospital's rule, if you like.)