L’Hôpital’s rule practice question 5

Will differentiating $e^{x}$ and $e^{-x}$ ever make this quotient simpler? As $x \to \infty$, which terms of the top and the bottom grow, and which fade away?

The form is $\tfrac{\infty}{\infty}$. Apply the rule twice and compare the result with the limit you started from.

Instead, divide the top and the bottom by the fastest-growing term, $e^{x}$, and use $e^{-2x} \to 0$.