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?

Hint What happens if you repeat the L'Hôpital's rule $p$ times?
