End behavior and horizontal asymptote of a rational function

Divide numerator and denominator by $x^3$, the highest power of $x$ appearing in the denominator: $h(x) = \frac{7x+2}{3x^3-x^2+1} = \frac{\frac{7}{x^2} + \frac{2}{x^3}}{3 - \frac{1}{x} + \frac{1}{x^3}}.$ As $x \to \pm\infty$, every term with $x$ in the denominator tends to $0$.

This leaves $\frac{0+0}{3-0+0} = 0$, so $\lim_{x\to\infty} h(x) = 0$ and $\lim_{x\to -\infty} h(x) = 0$. Since the degree of the numerator (1) is less than the degree of the denominator (3), this matches the general rule that such rational functions tend to 0 at both ends. Therefore the horizontal asymptote is $y = 0$.