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.