AP Calculus AB and BC glossary

One-sided limit

Also called: Left-hand limit, Right-hand limit

A one-sided limit is the value a function approaches as the input comes in from one direction only. The left-hand limit uses inputs below the point and the right-hand limit uses inputs above it. The two-sided limit exists exactly when both agree.

\lim_{x \to a^-} f(x) \quad \text{and} \quad \lim_{x \to a^+} f(x)

The notation $x \to a^-$ means $x$ approaches $a$ through values less than $a$ , and $x \to a^+$ means through values greater than $a$ . The superscript describes which side of the number line you are on, not the sign of the function.

One-sided limits are the tool for piecewise functions, jump discontinuities, and any question about behavior at an endpoint of a domain. They are also how you check differentiability at a corner.

The mistake

Reading the minus in $a^-$ as a negative number. In $\lim_{x \to 3^-}$ the inputs are values like 2.9 and 2.99, all positive, all just below 3.

Appears in: Unit 1: Limits and Continuity