Can I apply L'Hospital's rule to $\lim\limits_{x\to\infty} \frac{\sqrt2πx}{e^x}$?

Question

I have the following limit

$$\lim\limits_{x\to\infty} \frac{\sqrt2πx}{e^x}$$ My question is: should I apply l'hôpital or directly tend to 0.

Answer 1

Yes you should apply L'Hospital,but you must apply it when $x \to + \infty$ because in this case you will have the undetermined form $\frac{\infty}{\infty}$.

So the limit will be zero.

From this you can understand that $e^x$ goes to infinity much more quickly than $x$.

But when $x \to - \infty$ we have that the limit is $-\infty$

So in this case you can compute the limit directly.