1. The problem is to find the limit of the function $e^{-x}$ as $x$ approaches infinity. 2. The function is $f(x) = e^{-x}$. 3. Recall the rule: as $x \to \infty$, $e^{-x} = \frac{1}{e^x}$. 4. Since $e^x$ grows without bound as $x \to \infty$, $\frac{1}{e^x}$ approaches 0. 5. Therefore, $$\lim_{x \to \infty} e^{-x} = 0.$$