Prove that all values satisfy this expression I'm trying to find for what values for $p$ will cause $$\displaystyle\lim_{n\to\infty} \frac{\ln^p{n}}{n} = 0$$ I believe that one criteria for a limit approaching zero is that the top should be going to infinity slower than the bottom; however in this expression, it seems that I can make the top grows as fast as I can by adjusting $p$.
I tried to plot the graph of
$$
f(p) = \displaystyle\lim_{n\to\infty} \frac{\ln^p{n}}{n}
$$
and indeed, $\forall p$ yields $0$.
I was thinking if I can prove it without plotting every value. This is what I did:
$$
\begin{align}
&\displaystyle\lim_{n\to\infty} \frac{\ln^p{n}}{n} \\
\overset{L}{=}& \displaystyle\lim_{n\to\infty} \frac{p\ln{(n)}^{p-1}}{n} \\
\end{align}
$$
But then I got stuck here. It seems like this is not the way of solving it. But if this is not, then what is the way of solving this?

For $p>0$ and $q>0$, we have that
$$
\lim_{x\to\infty}\frac{\log^px}{x^q}
 =\lim_{x\to\infty}\Bigl(\frac{\log x}{x^{q/p}}\Bigr)^p
 =\Bigl(\lim_{x\to\infty}\frac{\log x}{x^{q/p}}\Bigr)^p
 =\Bigl(\lim_{x\to\infty}\frac1{(q/p)x^{q/p}}\Bigr)^p=0
$$
using continuity and l'Hôpital's rule.
