AP Calculus AB and BC glossary

Limit at infinity

Also called: End behavior

A limit at infinity is the value a function approaches as its input grows without bound in the positive or negative direction. It describes the end behavior of the graph, and a finite limit at infinity is exactly what produces a horizontal asymptote.

\lim_{x \to \infty} f(x) = L

For a rational function, compare degrees. If the numerator degree is smaller, the limit is 0. If the degrees are equal, the limit is the ratio of leading coefficients. If the numerator degree is larger, the function grows without bound and there is no finite limit.

The reliable general method is to divide the numerator and the denominator by the highest power of $x$ in the denominator, then use the fact that $\frac{1}{x^n} \to 0$ as $x \to \infty$ for any positive $n$ .

Watch the negative direction

As $x \to -\infty$ , $\sqrt{x^2}$ equals $|x| = -x$ , not $x$ . Forgetting this flips the sign of the answer and is the single most common error on end-behavior problems involving square roots.

Appears in: Unit 1: Limits and Continuity