End behavior and horizontal asymptote of a rational function

Since the numerator and denominator are both degree 3 polynomials, divide every term by $x^3$: $h(x) = \frac{x^3 - 8}{x^3 + 8} = \frac{1 - \frac{8}{x^3}}{1 + \frac{8}{x^3}}.$ As $x \to \infty$ or $x \to -\infty$, the terms $\frac{8}{x^3} \to 0$, so the expression approaches $\frac{1 - 0}{1 + 0} = 1$.

Since both one-sided limits at infinity equal the ratio of the leading coefficients ($1/1 = 1$), we get $\lim_{x\to\infty} h(x) = 1, \qquad \lim_{x\to -\infty} h(x) = 1.$ Because both limits are the same finite value, the function has a single horizontal asymptote at $y = 1$.