Find the horizontal and vertical asymptotes of​ f(x).
​f(x)equalsStartFraction 2 x Over x plus 4 EndFraction
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Part 1
Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice.
A.
The horizontal​ asymptote(s) can be described by the​ line(s)

enter your response here.
​(Type an equation. Use a comma to separate answers as​ needed.)
B.
There are no horizontal asymptotes

To find the horizontal asymptote(s) of the function \( f(x) = \frac{2x}{x + 4} \), follow these steps:

### Step 1: Analyze the degrees of the numerator and denominator.
- The degree of the numerator (\(2x\)) is 1.
- The degree of the denominator (\(x + 4\)) is 1.

### Step 2: Determine the horizontal asymptote based on the degrees.
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

For \( f(x) = \frac{2x}{x + 4} \), the leading coefficient in the numerator is 2, and the leading coefficient in the denominator is 1. Thus, the horizontal asymptote is:

\[
y = \frac{2}{1} = 2
\]

### Conclusion:
The horizontal asymptote is \( y = 2 \).

**Choice A** is the correct answer:
- The horizontal asymptote(s) can be described by the line(s) \( y = 2 \).

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Would you like to discuss the vertical asymptotes next? Here are five related questions:

1. How do you find vertical asymptotes for rational functions?
2. What happens to \( f(x) \) as \( x \) approaches the vertical asymptote?
3. Can a function have both horizontal and vertical asymptotes?
4. What if the degrees of the numerator and denominator were different?
5. How do horizontal asymptotes affect the end behavior of a function?

**Tip:** Horizontal asymptotes represent the value that a function approaches as \( x \) goes to positive or negative infinity.

Learn how to find horizontal asymptotes of the rational function f(x) = 2x / (x + 4) using the degree comparison method. Understand the concept of horizontal asymptotes and their significance in the behavior of rational functions. Suitable for students in Grades 11-12.

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