# 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.

• From the Taylor series, $$e^x \ge \frac{x^4}{4!}.$$
– user296602
Oct 11, 2017 at 3:42
• Have you learnt L'Hospital's rule? Oct 11, 2017 at 3:42
• @MonkeyKing yes, I kept getting infinty over infinty Oct 11, 2017 at 3:43

$$\lim_{x \to \infty} \frac{x^e}{e^x}= \lim_{x \to \infty} \frac{ex^{e-1}}{e^x}=e(e-1)(e-2)\lim_{x \to \infty} \frac{1}{x^{3-e}e^x}=0$$
If $x>0$ then $e^x>\frac{x^3}{6}$, by the power series for $e^x$.
So $\frac{x^e}{e^x}<\frac{6}{x^{3-e}}\to 0$ as $x\to\infty$.