1. The problem is to evaluate the expression $e^{-\infty^2}$. 2. Recall that $\infty$ represents an infinitely large number, and squaring it, $\infty^2$, is still infinitely large. 3. The expression becomes $e^{-\infty}$, which means the exponential function with a very large negative exponent. 4. The exponential function $e^x$ approaches 0 as $x$ approaches negative infinity. 5. Therefore, $e^{-\infty} = 0$. 6. So, the value of $e^{-\infty^2}$ is 0.