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.