1. The problem is to understand why $e^{-\infty} = 0$. 2. Recall the exponential function $e^x$ where $e$ is approximately 2.71828. 3. When the exponent $x$ is a very large negative number, say $-\infty$, it means we are looking at $e$ raised to a very large negative power. 4. The rule for exponents states that $e^{-a} = \frac{1}{e^a}$ for any positive number $a$. 5. So, $e^{-\infty} = \frac{1}{e^{\infty}}$. 6. Since $e^{\infty}$ means $e$ raised to an infinitely large positive number, it grows without bound and tends to infinity. 7. Therefore, $\frac{1}{e^{\infty}} = \frac{1}{\infty} = 0$. 8. In simple terms, as the exponent becomes very large and negative, the value of $e$ raised to that power gets closer and closer to zero. Final answer: $$e^{-\infty} = 0$$