Hi!

I need to find the limit of the following:

$\displaystyle \lim_{x\rightarrow\infty}\frac{x^k}{e^x}$ while $\displaystyle k\in\mathbb{N}$.

I know it's 0 since exponential functions grows faster then any polynomial functions, but I couldn't find a way to prove it.

$\displaystyle \lim_{x\rightarrow\infty}\frac{x^\ln x}{\ln ^xx}$

Please don't post a solution, I just need guidance and hints on what should I try.

Thanks in advanced!