One-sided limit

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above. In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above. The limit as x decreases in value approaching a (x approaches a 'from the right' or 'from above') can be denoted: The limit as x increases in value approaching a (x approaches a 'from the left' or 'from below') can be denoted: In probability theory it is common to use the short notation: The two one-sided limits exist and are equal if the limit of f(x) as x approaches a exists. In some cases in which the limit does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a 'two-sided limit'. In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.