When we found a limit, we sneaked up on c from both directions and the outputs met at one value. But sometimes the approach from the left and the approach from the right tell different stories. To talk about each separately, we use one-sided limits.

• The left-hand limit — approach with x a little below c: \lim_{x \to c^-} f(x).
• The right-hand limit — approach with x a little above c: \lim_{x \to c^+} f(x).

The little {}^- and {}^+ are like arrows pointing at c from each side.

A step in the road

Picture a function that jumps at x = 2: it sits at height 1 just to the left, then leaps up to height 3 just to the right. Walk a point in from each side and read where it lands.

From the left the point heads for 1; from the right it heads for 3: \lim_{x \to 2^-} f(x) = 1, \qquad \lim_{x \to 2^+} f(x) = 3. Two different answers — one for each side.

The two-sided limit needs agreement

Here is the rule that ties it together. The ordinary (two-sided) limit \lim_{x \to c} f(x) exists only when both one-sided limits exist and are equal:

For our jump, the sides disagree (1 \ne 3), so the two-sided limit at x = 2 does not exist — even though each one-sided limit is perfectly fine on its own.

Watch Sal read them off a graph