Always stick to the speed LIMIT

$\lim\limits_{x\to\infty}xe^{-x} = \lim\limits_{x\to\infty}\frac{x}{e^{x}}$

If we plug $\infty$ in here, we get a $\cfrac{\infty}{\infty}$ situation. From here we can go 2 ways:

Method 1: We can argue that $e^{x}$ as grows quicker as $x \to \infty$ than just $x$ . Since $e^{x}$ is in the denominator, we can argue that this limit will tend to $\boxed{0}$ (Also, this way of thinking works well with the pun i put for the title :D )

Method 2: We can also use L'Hopital's Rule here and differentiate the top and bottom: $\begin{aligned} \lim\limits_{x\to\infty}\frac{x}{e^{x}} \Longrightarrow \lim\limits_{x\to\infty}\frac{\frac{d}{dx}(x)}{\frac{d}{dx}(e^{x})} &= \lim\limits_{x\to\infty}\frac{1}{e^{x}} \\ &= \frac{1}{e^{\infty}} \\ &= \frac{1}{\infty} = \boxed{0} \end{aligned}$