the function r(X) = x+2/2x-3 has a horizontal asymptote of y = 

To determine the horizontal asymptote of the function \( r(x) = \frac{x + 2}{2x - 3} \), we need to analyze the behavior of the function as \( x \) approaches infinity or negative infinity.

### Step 1: Examine the degrees of the numerator and denominator
The given function is:
\[
r(x) = \frac{x + 2}{2x - 3}
\]
The degree of the numerator (\(x + 2\)) is 1, and the degree of the denominator (\(2x - 3\)) is also 1.

### Step 2: Identify the horizontal asymptote
When the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 1 (from \(x\)), and the leading coefficient of the denominator is 2 (from \(2x\)).

So, the horizontal asymptote is:
\[
y = \frac{1}{2}
\]

### Conclusion
The horizontal asymptote of the function \( r(x) = \frac{x + 2}{2x - 3} \) is \( y = \frac{1}{2} \).

Do you need more details or have any questions?

Here are some related questions:
1. What is the vertical asymptote of the function \( r(x) = \frac{x + 2}{2x - 3} \)?
2. How do you find the x-intercept of \( r(x) \)?
3. What happens to \( r(x) \) as \( x \) approaches the vertical asymptote?
4. Can you sketch the graph of \( r(x) = \frac{x + 2}{2x - 3} \)?
5. How does the horizontal asymptote change if the function were \( r(x) = \frac{2x + 5}{2x - 3} \)?

**Tip:** Horizontal asymptotes tell you the value that a function approaches as \( x \) becomes very large (positive or negative).

Learn how to determine the horizontal asymptote of the rational function r(x) = (x + 2) / (2x - 3) using leading coefficients. Understand the behavior of rational functions and their asymptotic properties. Suitable for advanced high school students.

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