In calculus, a one-sided limit refers to either one of the two limits of a function ${\displaystyle f(x)}$ of a real variable ${\displaystyle x}$ as ${\displaystyle x}$ approaches a specified point either from the left or from the right.

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

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

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

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

If the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists then the limits from the left and from the right both exist and are equal. 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 ${\displaystyle x}$ approaches ${\displaystyle a}$ is sometimes called a "two-sided limit".^{[citation needed]}

It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist.

If ${\displaystyle I}$ represents some interval that is contained in the domain of ${\displaystyle f}$ and if ${\displaystyle a}$ is a point in ${\displaystyle I}$ then the right-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle R}$ that satisfies:^{[verification needed]} $${\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<x-a<\delta {\text{ then }}|f(x)-R|<\varepsilon ,}$$ and the left-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle L}$ that satisfies: $${\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<a-x<\delta {\text{ then }}|f(x)-L|<\varepsilon .}$$