The expression $e^{-\infty}$ represents the mathematical limit of the exponential function as the exponent approaches negative infinity.

In this case:

$\lim_{x \to -\infty} e^x = 0$

So, as the exponent becomes more and more negative, $e^x$ approaches 0. Thus:

$e^{-\infty} = 0$

Key Insight:

The exponential function $e^x$ decays rapidly to zero as $x$ becomes very negative.

Would you like more details or have any questions about this concept?

Here are 5 related questions for further exploration:

1. What is the general behavior of $e^x$ as $x \to \infty$?
2. How does the graph of $e^x$ look for negative values of $x$?
3. What is the limit of $e^x$ as $x \to 0$?
4. How does the function $e^{-x}$ behave as $x \to \infty$?
5. What is the relationship between the function $e^x$ and the natural logarithm?

Tip: Exponential functions like $e^x$ are important in growth/decay models and can help understand phenomena in fields like biology, economics, and physics!