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?

Solution 1:

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.

Solution 2:

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