# 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 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.