y= (e^x)/((e^x)-2) Find vertical and the horizontal asymptote

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.

y = (e^x)/((e^x)-2) Find vertical and the horizontal asymptote

Explore how to find the vertical and horizontal asymptotes for the function y = (e^x)/(e^x - 2). This solution involves analyzing exponential functions and limits, ideal for students in Grades 10-12.

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