L’Hôpital’s rule practice question 8

As $x \to \infty$, which term of the top and which of the bottom grows fastest? Read $e^{-x}$ carefully before you decide.

The form is $\tfrac{\infty}{\infty}$. You can apply the rule twice, checking the form before each, or compare leading terms.

Divide the top and the bottom by $x^{2}$, using $\dfrac{\log x}{x^{2}} \to 0$ and $\dfrac{e^{-x}}{x^{2}} \to 0$.