We are tasked with finding the limit of the expression:

$\lim_{x \to \infty} 4e^{-4x}$

Step-by-step explanation:

1. The given function is $4e^{-4x}$.

2. As $x \to \infty$, $e^{-4x}$ approaches zero because the exponential function with a negative exponent tends toward zero as $x$ grows larger.

   $\lim_{x \to \infty} e^{-4x} = 0$

3. Multiplying this by 4:

   $4 \times 0 = 0$

Thus, the limit is:

$\boxed{0}$

This limit exists and equals 0.

Would you like further details on exponential decay or any other part of the explanation?

Related questions:

1. What is the behavior of exponential functions as $x \to -\infty$?
2. How does the constant factor (like 4 in this case) affect the limit of an exponential function?
3. What are the properties of exponential decay in terms of real-world applications?
4. How would the limit change if the exponent were positive (e.g., $e^{4x}$)?
5. What is the difference between exponential growth and decay in terms of limits?

Tip: Exponential decay functions like $e^{-kx}$ (for positive $k$) always approach zero as $x \to \infty$.