AP Calculus AB and BC glossary

Infinite limit

An infinite limit means the function grows without bound as the input approaches a point, written as equal to positive or negative infinity. Because infinity is not a number, an infinite limit is a description of how the limit fails to exist, not a value.

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

Writing $\lim_{x \to a} f(x) = \infty$ is shorthand for a specific kind of failure: the outputs exceed every bound you name. It tells you more than simply saying the limit does not exist, which is why the notation is worth keeping.

An infinite limit at $x = a$ means the line $x = a$ is a vertical asymptote. Check each side separately: a function can go to $+\infty$ on one side and $-\infty$ on the other, as $\frac{1}{x}$ does at 0.

Appears in: Unit 1: Limits and Continuity