AP Calculus AB and BC glossary

Infinite discontinuity

Also called: Essential discontinuity

An infinite discontinuity occurs where at least one one-sided limit is infinite, so the function grows without bound near the point. It always corresponds to a vertical asymptote and can never be removed by redefining a value.

These arise when a denominator approaches zero and the numerator does not. In $\frac{1}{(x-3)^2}$ the function goes to $+\infty$ on both sides of 3; in $\frac{1}{x-3}$ it goes to $-\infty$ from the left and $+\infty$ from the right.

Check before concluding

A zero denominator does not guarantee an infinite discontinuity. If the numerator shares the factor, the discontinuity may be removable instead, so always factor first.

Appears in: Unit 1: Limits and Continuity