# Help with evaluating limits

• March 9th 2013, 01:51 AM
Stormey
Help with evaluating limits
Hi!

I need to find the limit of the following:

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

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

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

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

• March 9th 2013, 02:05 AM
Prove It
Re: Help with evaluating limits
The first one can be proven using L'Hospital's Rule "k" times.
• March 9th 2013, 02:38 AM
Stormey
Re: Help with evaluating limits
Hi Prove It!
Thanks for the help.

That's brilliant.
$\lim_{x\rightarrow \infty}\frac{x^k}{e^x}=\lim_{x\rightarrow \infty}\frac{kx^{k-1}}{e^x}=\lim_{x\rightarrow \infty}\frac{k(k-1)x^{k-2}}{e^x}=...=\lim_{x\rightarrow \infty}\frac{k!}{e^x}=0$
(correct me if there's an error)