# 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:

${\displaystyle \lim _{x\to a^{+}}f(x)\ }$ or ${\displaystyle \lim _{x\downarrow a}\,f(x)}$ or ${\displaystyle \lim _{x\searrow a}\,f(x)}$ or ${\displaystyle \lim _{x{\underset {>}{\to }}a}f(x)}$

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

${\displaystyle \lim _{x\to a^{-}}f(x)\ }$ or ${\displaystyle \lim _{x\uparrow a}\,f(x)}$ or ${\displaystyle \lim _{x\nearrow a}\,f(x)}$ or ${\displaystyle \lim _{x{\underset {<}{\to }}a}f(x)}$

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

${\displaystyle \lim _{x\to a}f(x)\,}$

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:

${\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0

Similarly, the left-sided limit can be rigorously defined as:

${\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0

Where ${\displaystyle I}$ represents some interval that is within the domain of ${\displaystyle f}$

## Examples

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

${\displaystyle \lim _{x\rightarrow 0^{+}}{1 \over 1+2^{-1/x}}=1,}$

whereas

${\displaystyle \lim _{x\rightarrow 0^{-}}{1 \over 1+2^{-1/x}}=0.}$

## 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.