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 the left or from the right.

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

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

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

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

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

${\displaystyle f(x-)}$ for the left limit and ${\displaystyle f(x+)}$ 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

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

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

${\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<x-a<\delta \Rightarrow |f(x)-L|<\varepsilon ),}$

and the left-sided limit can be rigorously defined as

${\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<a-x<\delta \Rightarrow |f(x)-L|<\varepsilon ),}$

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