𝑦 = (3𝑒^4𝑥)/(1-e^4x)
Find the horizontal asymptotes

To find the horizontal asymptotes of the function \( y = \frac{3e^{4x}}{1 - e^{4x}} \), we need to analyze the behavior of the function as \( x \) approaches infinity (\( x \to \infty \)) and as \( x \) approaches negative infinity (\( x \to -\infty \)).

### Step 1: Analyze the limit as \( x \to \infty \)

As \( x \) approaches infinity:

- \( e^{4x} \) becomes very large.
- Thus, \( 1 - e^{4x} \) approaches a large negative value.

The function simplifies to:

\[
y \approx \frac{3e^{4x}}{-e^{4x}} = \frac{3}{-1} = -3
\]

So, as \( x \to \infty \), \( y \) approaches \( -3 \).

### Step 2: Analyze the limit as \( x \to -\infty \)

As \( x \) approaches negative infinity:

- \( e^{4x} \) approaches zero.

The function simplifies to:

\[
y \approx \frac{3 \cdot 0}{1 - 0} = 0
\]

So, as \( x \to -\infty \), \( y \) approaches \( 0 \).

### Conclusion

The horizontal asymptotes are:
- \( y = -3 \) as \( x \to \infty \)
- \( y = 0 \) as \( x \to -\infty \)

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How do you find vertical asymptotes for rational functions?
2. What is the significance of horizontal asymptotes in the context of a graph?
3. How does the behavior of exponential functions affect asymptotes?
4. Can a function have more than two horizontal asymptotes?
5. How do you differentiate between horizontal and slant asymptotes?

**Tip:** When analyzing horizontal asymptotes, always consider the behavior of the function as \( x \) approaches both positive and negative infinity.

Learn how to find the horizontal asymptotes of the function 𝑦 = (3𝑒^4𝑥)/(1-e^4x) with a step-by-step explanation. Understand the behavior of exponential functions as 𝑥 approaches infinity and negative infinity. Suitable for advanced high school students.

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