One-Sided Limit

What is a One-sided Limit?

A one-sided limit is either of the two limits of a function of a variable as the variable approaches a specified point from either above or below. The picture below graphs out the two one-sided limits of a function.

Image from mathwarehouse.com


The limit of a function f(x) of a real variable x is the value l which f(x) approaches as x approaches some value x0. Formally:



Sometimes f(x) approaches different values when x approaches x0 from different directions; this happens in the case of a jump discontinuity. [Steal the image we use for Jump Discontinuity, and re-use it here.] In this case, f(x) can't be said to have a unique limit at x0, but it may well have two one-sided limits: one as x approaches from the left, and one as x approaches from the right.


One-sided vs. Two Sided
Two-sided limits exist and are equal if the limit of a function of a variable as it approaches a specified point exists. Occasionally, one of the two-sided limits exists and the other does not, or sometimes neither limit exists. Referring to limits as two-sides distinguishes it from the one-sided limits from the right and/or the left.