To find the horizontal asymptote of the function \( f(x) = \frac{2x + 3}{x - 1} \), we analyze its behavior as \( x \) approaches infinity or negative infinity.

### Step-by-Step Process:
1. **Rewrite the function:**
   \[
   f(x) = \frac{2x + 3}{x - 1}
   \]
   
2. **Divide the numerator and denominator by \( x \), the highest power of \( x \):**
   \[
   f(x) = \frac{\frac{2x + 3}{x}}{\frac{x - 1}{x}} = \frac{2 + \frac{3}{x}}{1 - \frac{1}{x}}
   \]

3. **Evaluate the limits as \( x \to \infty \) or \( x \to -\infty \):**
   - As \( x \to \infty \), \( \frac{3}{x} \to 0 \) and \( \frac{1}{x} \to 0 \). Therefore, the function approaches:
     \[
     f(x) \to \frac{2 + 0}{1 - 0} = 2
     \]
   - Similarly, as \( x \to -\infty \), the function also approaches:
     \[
     f(x) \to \frac{2 + 0}{1 - 0} = 2
     \]

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

Would you like further details on horizontal asymptotes or have any questions?

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Here are 5 related questions:
1. How do you find vertical asymptotes for rational functions?
2. What happens to \( f(x) \) at its vertical asymptote?
3. Can a function have more than one horizontal asymptote?
4. How do you find slant asymptotes for functions with different degrees?
5. What is the difference between horizontal and oblique asymptotes?

**Tip:** For rational functions, the horizontal asymptote depends on the degree of the numerator and denominator.

Discover how to find the horizontal asymptote of the rational function f(x) = (2x+3)/(x-1) with a detailed solution. This example demonstrates key algebraic techniques such as dividing by the highest degree of x and evaluating limits as x approaches infinity. Suitable for high school students in Grades 10-12.

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