one-sided limit

Picture standing at a point on a number line and asking, 'where is the function heading?' — but you are allowed to approach from only one direction. If you sneak up on a from the left, using inputs smaller than a, you get the left-hand limit. If you come in from the right, using inputs larger than a, you get the right-hand limit. They are two separate questions, and the answers can genuinely differ, like two people meeting at a doorway from opposite hallways.

We write lim x->a^- f(x) for the left-hand limit (x approaches a through values below a) and lim x->a^+ f(x) for the right-hand limit (x approaches through values above a). Here is the key fact that ties them to the ordinary limit: the two-sided limit lim x->a f(x) exists and equals L if and only if both one-sided limits exist and are equal to that same L. In symbols, the limit exists exactly when lim x->a^- f(x) = lim x->a^+ f(x). If the two sides disagree, the ordinary limit does not exist.

One-sided limits are the natural tool for functions that change their rule at a point — like a tax bracket, a step in a price chart, or a piecewise definition. They are also why we can speak of limits at the very edge of a domain, where there is only one side to approach from. A 'jump discontinuity' is precisely the case where both one-sided limits exist but are unequal.