How to evaluate $\lim_{x \to \infty} \frac{x^x}{e^x x!}$? How should I go about evaluating this limit?
$$ \lim_{x \to \infty} \frac{x^x}{e^x x!} $$
Intuitively I am unsure whether this tends towards infinity or $0$. So far I have tried using L'Hopital however that only makes the expression worse. Take the logarithm seemed promising but I was unsure how to continue that solution.

As Thomas said, using Stirling’s approximation:
$$x!\sim\sqrt{2\pi x}\left(\frac xe\right)^x$$
giving:
$$\frac{1}{x!}\left(\frac xe\right)^x\sim\frac{1}{\sqrt{2\pi x}}\stackrel{x\to\infty^+}{\to}0$$
