Limits Involving Infinity and Asymptotic Behavior

Key rules

• $\lim_{x \to a} f(x) = \pm\infty$ means $f(x)$ grows without bound near $a$. The limit still does not exist as a number; $\pm\infty$ describes how it fails.
• Vertical asymptote $x = a$: at least one one-sided limit at $a$ is $\pm\infty$. For $\frac{p(x)}{q(x)}$: $q(a) = 0$ and $p(a) \neq 0$. Find the sign on each side with a sign check.
• Horizontal asymptote $y = L$: $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$. Check both directions; there can be two.
• Rational functions at $\pm\infty$: say $\frac{\infty}{\infty}$ (indeterminate form), then divide by the highest power in the denominator. Degree top < bottom: $0$; equal: ratio of leading coefficients; top > bottom: $\pm\infty$.
• Square roots at $-\infty$: $\sqrt{x^2} = |x| = -x$ when $x < 0$. For $\infty - \infty$ with roots, multiply by the conjugate.