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. One should write either:

for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly

for the limit as x increases in value approaching a (x approaches a "from the left" or "from below").

The two one-sided limits exist and are equal if and only 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.

Examples

One example of a function with different one-sided limits is the following:

whereas

Relation to topological definition of limit

The one-sided limit to a point p corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p.

Abel's theorem

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.

See also

• Real projective line

External links

• One-sided limit on PlanetMath