one-sided limit

The value a function approaches from only one direction — the left or the right.

The left-hand limit $\displaystyle\lim_{x\to a^-} f(x)$ only considers $x$ approaching $a$ from below; the right-hand limit $\displaystyle\lim_{x\to a^+} f(x)$ only from above. The two-sided limit exists exactly when both one-sided limits exist and agree.

Examples

For a step function jumping at $x=0$ , the left- and right-hand limits at $0$ differ, so the two-sided limit fails to exist.