Help with evaluating limits

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!

Re: Help with evaluating limits

The first one can be proven using L'Hospital's Rule "k" times.

Re: Help with evaluating limits

Hi Prove It!

Thanks for the help.

That's brilliant.

didn't think about it.

$\displaystyle \lim_{x\rightarrow \infty}\frac{x^k}{e^x}=\lim_{x\rightarrow \infty}\frac{kx^{k-1}}{e^x}=\lim_{x\rightarrow \infty}\frac{k(k-1)x^{k-2}}{e^x}=...=\lim_{x\rightarrow \infty}\frac{k!}{e^x}=0$

(correct me if there's an error)

what about the second one?

any thoughts? (Itwasntme)