TITLE: Limit of $\frac{(x^x)}{(x!)}$ as $x$ approaches infinity
QUESTION [5 upvotes]: So, I'm learning limits right now in calculus class.
When $x$ approaches infinity, what does this expression approach?
$$\frac{(x^x)}{(x!)}$$
Why?  Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?

REPLY [8 votes]: Hint
$$\frac{x^x}{x!} > \frac{x}{1}$$