One-sided limit


In calculus, a one-sided limit is either of the two limits of a function f of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a can be denoted:
The limit as x increases in value approaching a 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 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.
The right-sided limit can be rigorously defined as
and the left-sided limit can be rigorously defined as
where represents some interval that is within the domain of.

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. Alternatively, one may consider the domain with a half-open interval topology.

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.