AP Calculus AB and BC glossary

Right-hand limit

Also called: Right limit, Right-side limit

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

The notation $x \to a^+$ means $x$ approaches $a$ through inputs larger than $a$ , values like $3.1$ and $3.01$ closing in on $3$ . On a graph you trace the curve toward the point from the right; on a piecewise function you evaluate the branch that governs inputs above $a$ .

The right-hand limit is the whole story at the left end of a domain, where inputs below the point do not exist. For $f(x) = \sqrt{x}$ the only limit at $0$ is $\lim_{x \to 0^+} \sqrt{x} = 0$ . Pairing it with the left-hand limit is how you confirm a two-sided limit exists.

The mistake

Deciding a limit fails just because only one side exists. At the left edge of a domain the two-sided limit is undefined, yet the right-hand limit can be perfectly well defined and is exactly what the graph shows.

Appears in: Unit 1: Limits and Continuity