Let's re-evaluate the asymptotes for the given function \( f(x) = \frac{7e^x}{e^x - 6} \).

### Horizontal Asymptotes
To find the horizontal asymptotes, we analyze the behavior of the function as \( x \) approaches \( \infty \) and \( -\infty \).

**As \( x \to \infty \):**
\[ \lim_{x \to \infty} \frac{7e^x}{e^x - 6} \]
Divide the numerator and the denominator by \( e^x \):
\[ \lim_{x \to \infty} \frac{7e^x / e^x}{(e^x - 6) / e^x} = \lim_{x \to \infty} \frac{7}{1 - 6 / e^x} \]
As \( x \to \infty \), \( \frac{6}{e^x} \to 0 \), so the limit is:
\[ \frac{7}{1 - 0} = 7 \]
Thus, there is a horizontal asymptote at \( y = 7 \).

**As \( x \to -\infty \):**
\[ \lim_{x \to -\infty} \frac{7e^x}{e^x - 6} \]
As \( x \to -\infty \), \( e^x \to 0 \). Therefore:
\[ \lim_{x \to -\infty} \frac{7 \cdot 0}{0 - 6} = \frac{0}{-6} = 0 \]
Thus, there is a horizontal asymptote at \( y = 0 \).

### Vertical Asymptotes
To find vertical asymptotes, we look for values of \( x \) where the denominator is zero and the numerator is non-zero.

**Setting the denominator equal to zero:**
\[ e^x - 6 = 0 \]
\[ e^x = 6 \]
Taking the natural logarithm of both sides:
\[ x = \ln(6) \]

At \( x = \ln(6) \), the function's denominator is zero, which suggests a vertical asymptote at this point. We need to ensure the numerator does not also go to zero here:
\[ 7e^x \neq 0 \text{ for any real } x \]

Therefore, there is a vertical asymptote at:
\[ x = \ln(6) \]

### Summary of Corrected Asymptotes

1. **Horizontal Asymptotes:**
   - \( y = 7 \)
   - \( y = 0 \)

2. **Vertical Asymptote:**
   - \( x = \ln(6) \)

Would you like more details or have any questions about these steps?

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Here are 5 questions you might ask next:
1. How do you determine if a limit results in an indeterminate form?
2. Can you explain the concept of horizontal and vertical asymptotes further?
3. How do you find limits as \( x \) approaches infinity or negative infinity?
4. What are some techniques for solving exponential equations?
5. Can you graph the function \( f(x) = \frac{7e^x}{e^x - 6} \)?

**Tip:** When dealing with exponential functions, always check the behavior at infinity and negative infinity to accurately determine the horizontal asymptotes.

Explore the horizontal and vertical asymptotes of the function f(x) = (7e^x) / (e^x - 6). Learn how to determine asymptotic behavior as x approaches infinity and negative infinity. Suitable for advanced high school students and anyone interested in limits and exponential functions.

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