# One-Sided Limits

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 the left or from the right.

The limit as *x* decreases in value approaching *a* (*x* approaches *a* or "from above") can be denoted:

- or or or

The limit as *x* increases in value approaching *a* (*x* approaches *a* or "from below") can be denoted:

- or or or

In probability theory it is common to use the short notation:

- for the left limit and for the right limit.

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

and the left-sided limit can be rigorously defined as

where I represents some interval that is within the domain of f.

## Contents

## Examples

One example of a function with different one-sided limits is the following (cf. picture):

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.

## Resources

- One-sided limits from graphs, Khan Academy

## Licensing

Content obtained and/or adapted from:

- One-sided limit, Wikipedia under a CC BY-SA license