https://file.aiquickdraw.com/mathbot/file/1727303966507nroix664.jpg

The equation in your image is:

\[
y = \frac{2e^x}{e^x - 5}
\]

The instruction below seems to ask for the **vertical and horizontal asymptotes** of this function. Let’s determine those step by step.

### 1. Horizontal Asymptote:
- To find the horizontal asymptote, we need to analyze the behavior of the function as \(x\) approaches infinity (\(x \to \infty\)) and negative infinity (\(x \to -\infty\)).

For large \(x\):
\[
y = \frac{2e^x}{e^x - 5} \approx \frac{2e^x}{e^x} = 2
\]
As \(x \to \infty\), the horizontal asymptote is:
\[
y = 2
\]

For large negative \(x\) (\(x \to -\infty\)):
\[
e^x \to 0, \quad y = \frac{2e^x}{e^x - 5} \to \frac{0}{-5} = 0
\]
Thus, as \(x \to -\infty\), the horizontal asymptote is:
\[
y = 0
\]

### 2. Vertical Asymptote:
Vertical asymptotes occur when the denominator equals zero, as long as the numerator is not zero at the same point.

Set the denominator to zero:
\[
e^x - 5 = 0 \quad \Rightarrow \quad e^x = 5
\]
Taking the natural logarithm of both sides:
\[
x = \ln(5)
\]
Thus, there is a vertical asymptote at \(x = \ln(5)\).

### Summary:
- **Horizontal asymptotes**: \(y = 2\) as \(x \to \infty\), and \(y = 0\) as \(x \to -\infty\).
- **Vertical asymptote**: \(x = \ln(5)\).

Do you want further details or have any questions? 

### Related Questions:
1. How do we graph the function and its asymptotes?
2. What happens to the function near the vertical asymptote \(x = \ln(5)\)?
3. How do we confirm the horizontal asymptotes using limits?
4. Can you explain the behavior of the function around \(x = 0\)?
5. How do transformations affect the graph of \(y = \frac{e^x}{e^x - 5}\)?

### Tip:
For rational functions, vertical asymptotes are found where the denominator is zero, and horizontal asymptotes describe end behavior.

Find the vertical and horizontal asymptotes of the function y = (2e^x) / (e^x - 5).

Discover how to find the vertical and horizontal asymptotes of the exponential function y = (2e^x) / (e^x - 5). Explore detailed step-by-step methods covering limits and asymptotic behavior. This problem is ideal for students studying calculus and exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation