Let's analyze the function:

$y = \frac{e^x}{e^x - 2}$

1. Vertical Asymptote

A vertical asymptote occurs when the denominator of a function approaches zero, leading to the function approaching infinity.

The denominator is $e^x - 2$. Set it equal to zero to find the vertical asymptote:

$e^x - 2 = 0$

$e^x = 2$

Taking the natural logarithm on both sides:

$x = \ln(2)$

So, there is a vertical asymptote at:

$x = \ln(2)$

2. Horizontal Asymptote

To find the horizontal asymptote, examine the behavior of the function as $x \to \infty$ and $x \to -\infty$.

• As $x \to \infty$:

  $e^x$ grows very large, so $e^x - 2 \approx e^x$. Thus, the function behaves like:

  $y \approx \frac{e^x}{e^x} = 1$

  Therefore, as $x \to \infty$, $y \to 1$.

• As $x \to -\infty$:

  $e^x$ approaches 0, so the denominator $e^x - 2 \approx -2$. Thus, the function behaves like:

  $y \approx \frac{0}{-2} = 0$

  Therefore, as $x \to -\infty$, $y \to 0$.

Conclusion:

• Vertical asymptote: $x = \ln(2)$
• Horizontal asymptotes: $y = 1$ as $x \to \infty$, and $y = 0$ as $x \to -\infty$

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

1. How do you find the horizontal asymptote for a rational function?
2. What happens to the function near $x = \ln(2)$?
3. Why does $e^x$ grow faster than any polynomial as $x \to \infty$?
4. How do we derive natural logarithms from exponential equations?
5. What are oblique asymptotes, and when do they occur?

Tip: When dealing with exponential functions, analyzing the behavior at extreme values of $x$ helps in identifying asymptotes easily.