End behavior and horizontal asymptote of a rational function

Since the numerator and denominator are both degree 4 polynomials, divide every term by $x^4$: $f(x) = \frac{2 - \frac{1}{x^2} + \frac{1}{x^4}}{1 + \frac{3}{x} - \frac{2}{x^4}}.$ As $x \to \infty$ or $x \to -\infty$, every term of the form $c/x^n$ (with $n>0$) approaches $0$, regardless of the sign of $x$.

This leaves $\lim_{x\to\infty} f(x) = \lim_{x\to-\infty} f(x) = \frac{2-0+0}{1+0-0} = 2.$ This matches the shortcut rule: when degrees are equal, the limit at infinity equals the ratio of leading coefficients, $2/1 = 2$. Since $f(x) \to 2$ in both directions, the graph has a single horizontal asymptote at $y = 2$.