AP Calculus AB and BC glossary

Left-hand limit

Also called: Left limit, Left-side limit

The left-hand limit is the value a function approaches as the input moves toward a point through values below it. It is written with a minus superscript on the target. The two-sided limit exists only when the left-hand and right-hand limits are equal.

The notation $x \to a^-$ means $x$ approaches $a$ through inputs smaller than $a$ , values like $2.9$ and $2.99$ closing in on $3$ . On a graph you trace the curve toward the point from the left; on a piecewise function you evaluate the branch that governs inputs below $a$ .

Left-hand limits are how you diagnose jump discontinuities, read behavior at a domain endpoint, and check differentiability at a corner. A two-sided limit exists only when the left-hand limit and the right-hand limit agree, so the left-hand value is half of every existence check.

The mistake

On a piecewise function, using the branch that owns the point $a$ to compute the left-hand limit. The left-hand limit only sees inputs below $a$ , so use the rule that applies for $x < a$ , even when a different rule defines $f(a)$ itself.

Appears in: Unit 1: Limits and Continuity