Horizontal asymptote of a rational function

Since the numerator and denominator are both degree 2, we divide every term by $x^2$: $f(x) = \frac{3 - \frac{5}{x} + \frac{1}{x^2}}{2 + \frac{1}{x} - \frac{4}{x^2}}.$ As $x \to \infty$, every term with $x$ in the denominator tends to 0.

This leaves $\lim_{x\to\infty} f(x) = \frac{3}{2},$ which is just the ratio of the leading coefficients. Therefore the graph has a horizontal asymptote at $y = \frac{3}{2}$.