calculus - Limit $lim_{x to infty}x^e/e^x$

I had this problem on my math test, and was stuck on it for quite some time.

$\lim_{x \to \infty}x^e/e^x$

I knew that the bottom grew faster than the top, but I didn't know how to prove it. I wrote that the limit approaches 0, but I am not sure how to prove it mathematically.

Answer

Show first that it is in indeterminate form.

Then perform L'Hopital's rule, differentiating the top and bottom.

$$\lim_{x \to \infty} \frac{x^e}{e^x}= \lim_{x \to \infty} \frac{ex^{e-1}}{e^x}=e(e-1)(e-2)\lim_{x \to \infty} \frac{1}{x^{3-e}e^x}=0$$