Estimate the limit numerically if it exists. (If an answer does not exist, enter DNE.)

lim *x*→∞ 4*e*−4*x*}

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 \( x \to \infty \).

Estimate the limit numerically if it exists. (If an answer does not exist, enter DNE.) lim x→∞ 4e−4x

Learn how to find the limit of the exponential decay function 4e^(-4x) as x approaches infinity. This problem explores key mathematical concepts such as limits and exponential decay, using the limit theorem for negative exponents. Suitable for high school students in Grades 10-12.

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