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:
- or or or
for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly
- or or or
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 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.
The right-sided limit can be rigorously defined as:
Similarly, 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.