TITLE: What is $\lim _{x\to \infty }\left(\frac{e^x}{x^n}\right)$?
QUESTION [8 upvotes]: What is $\lim _{x\to \infty }\left(\frac{e^x}{x^n}\right)$
?

REPLY [5 votes]: Recall that 
$$
e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}=1+\frac{x}{1!}+\frac{x^2}{2!}+\ldots+\frac{x^n}{n!}+\ldots
$$
then, for each fixed $n$, as $x \to +\infty$, you get
$$
e^x>\frac{x^{n+1}}{(n+1)!}
$$ and
$$
\frac{e^x}{x^n}>\frac{x}{(n+1)!} \longrightarrow +\infty, \quad\text{as} \quad x \to +\infty,
$$ and the desired limit is $ +\infty$.