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.^[1][2]

The limit as ${\displaystyle x}$ decreases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the right"^[3] or "from above") can be denoted:^[1][2]

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

The limit as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left"^[4][5] or "from below") can be denoted:^[1][2]

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

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

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

${\displaystyle x}$

${\displaystyle a}$

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.

Formal definition

Definition

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:^{[6][verification needed]}

$${\displaystyle {\text{for all ))\varepsilon >0\;{\text{ there exists some ))\delta >0\;{\text{ such that for all ))x\in I,{\text{ if ))\;0<x-a<\delta {\text{ then ))|f(x)-R|<\varepsilon ,}$$

${\displaystyle x}$

${\displaystyle a}$

${\displaystyle L}$

$${\displaystyle {\text{for all ))\varepsilon >0\;{\text{ there exists some ))\delta >0\;{\text{ such that for all ))x\in I,{\text{ if ))\;0<a-x<\delta {\text{ then ))|f(x)-L|<\varepsilon .}$$

We can represent the same thing more symbolically, as follows.

Let ${\displaystyle I}$ represent an interval, where ${\displaystyle I\subseteq \mathrm {domain} (f)}$, and ${\displaystyle a\in I}$.

$${\displaystyle \lim _{x\to a^{+))f(x)=R~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<x-a<\delta \longrightarrow |f(x)-R|<\varepsilon ))}$$

$${\displaystyle \lim _{x\to a^{-))f(x)=L~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<a-x<\delta \longrightarrow |f(x)-L|<\varepsilon ))}$$

Intuition

In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value.

For reference, the formal definition for the limit of a function at a point is as follows:

$${\displaystyle \lim _{x\to a}f(x)=L~~~\iff ~~~\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,0<|x-a|<\delta \implies |f(x)-L|<\varepsilon }$$

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between ${\displaystyle x}$ and ${\displaystyle a}$ is

$${\displaystyle |x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|}$$

For the limit from the right, we want ${\displaystyle x}$ to be to the right of ${\displaystyle a}$, which means that ${\displaystyle a<x}$, so ${\displaystyle x-a}$ is positive. From above, ${\displaystyle x-a}$ is the distance between ${\displaystyle x}$ and ${\displaystyle a}$. We want to bound this distance by our value of ${\displaystyle \delta }$, giving the inequality ${\displaystyle x-a<\delta }$. Putting together the inequalities ${\displaystyle 0<x-a}$ and ${\displaystyle x-a<\delta }$ and using the transitivity property of inequalities, we have the compound inequality ${\displaystyle 0<x-a<\delta }$.

Similarly, for the limit from the left, we want ${\displaystyle x}$ to be to the left of ${\displaystyle a}$, which means that ${\displaystyle x<a}$. In this case, it is ${\displaystyle a-x}$ that is positive and represents the distance between ${\displaystyle x}$ and ${\displaystyle a}$. Again, we want to bound this distance by our value of ${\displaystyle \delta }$, leading to the compound inequality ${\displaystyle 0<a-x<\delta }$.

Now, when our value of ${\displaystyle x}$ is in its desired interval, we expect that the value of ${\displaystyle f(x)}$ is also within its desired interval. The distance between ${\displaystyle f(x)}$ and ${\displaystyle L}$, the limiting value of the left sided limit, is ${\displaystyle |f(x)-L|}$. Similarly, the distance between ${\displaystyle f(x)}$ and ${\displaystyle R}$, the limiting value of the right sided limit, is ${\displaystyle |f(x)-R|}$. In both cases, we want to bound this distance by ${\displaystyle \varepsilon }$, so we get the following: ${\displaystyle |f(x)-L|<\varepsilon }$ for the left sided limit, and ${\displaystyle |f(x)-R|<\varepsilon }$ for the right sided limit.

Examples

Example 1: The limits from the left and from the right of ${\displaystyle g(x):=-{\frac {1}{x))}$ as ${\displaystyle x}$ approaches ${\displaystyle a:=0}$ are

$${\displaystyle \lim _{x\to 0^{-)){-1/x}=+\infty \qquad {\text{ and ))\qquad \lim _{x\to 0^{+)){-1/x}=-\infty }$$

${\displaystyle \lim _{x\to 0^{-)){-1/x}=+\infty }$

${\displaystyle x}$

${\displaystyle x\to 0^{-))$

${\displaystyle x\to 0}$

${\displaystyle x}$

${\displaystyle x<0}$

${\displaystyle -1/x}$

${\displaystyle \lim _{x\to 0^{-)){-1/x))$

${\displaystyle +\infty }$

${\displaystyle -\infty }$

${\displaystyle x}$

${\displaystyle 0}$

${\displaystyle \lim _{x\to 0^{+)){-1/x}=-\infty }$

${\displaystyle x}$

${\displaystyle x>0}$

${\displaystyle x}$

${\displaystyle x}$

${\displaystyle 0}$

${\displaystyle -1/x}$

${\displaystyle \lim _{x\to 0^{+)){-1/x))$

${\displaystyle -\infty .}$

Example 2: One example of a function with different one-sided limits is ${\displaystyle f(x)={\frac {1}{1+2^{-1/x))},}$ (cf. picture) where the limit from the left is ${\displaystyle \lim _{x\to 0^{-))f(x)=0}$ and the limit from the right is ${\displaystyle \lim _{x\to 0^{+))f(x)=1.}$ To calculate these limits, first show that

$${\displaystyle \lim _{x\to 0^{-))2^{-1/x}=\infty \qquad {\text{ and ))\qquad \lim _{x\to 0^{+))2^{-1/x}=0}$$

${\displaystyle \lim _{x\to 0^{-)){-1/x}=+\infty {\text{ and ))\lim _{x\to 0^{+)){-1/x}=-\infty }$

$${\displaystyle \lim _{x\to 0^{+)){\frac {1}{1+2^{-1/x))}={\frac {1}{1+\displaystyle \lim _{x\to 0^{+))2^{-1/x))}={\frac {1}{1+0))=1}$$

${\displaystyle \lim _{x\to 0^{-)){\frac {1}{1+2^{-1/x))}=0}$

${\displaystyle \lim _{x\to 0^{-))1+2^{-1/x}=\infty .}$

${\displaystyle \lim _{x\to 0^{-))f(x)\neq \lim _{x\to 0^{+))f(x),}$

${\displaystyle \lim _{x\to 0}f(x)}$

Relation to topological definition of limit

The one-sided limit to a point ${\displaystyle p}$ 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 ${\displaystyle p.}$^{[1][verification needed]} Alternatively, one may consider the domain with a half-open interval topology.^{[citation needed]}

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.^{[citation needed]}