One-sided limit

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.¹²

The limit, as ${\displaystyle x}$ decreases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the right"³ or "from above"), is denoted:¹²

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

The limit, as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left"⁴⁵ or "from below"), is denoted:¹²

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

If the limits from the left and right both exist and are equal, then the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists. Conversely, if the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists, then the limits from left and right both exist and are equal. Consequently, the limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ is sometimes called a "two-sided limit". It is denoted: $${\displaystyle \lim _{x\to a}f(x).}$$

In some cases in which the two-sided limit does not exist, the two individual one-sided limits nonetheless exist and they are then necessarily unequal.

It is possible for only one of the two one-sided limits to 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 a function ${\displaystyle f}$ 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

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

and the left-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle L}$ that satisfies:

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

These definitions can be represented more symbolically as follows: Let ${\displaystyle I}$ represent an interval, where ${\displaystyle I\subseteq \mathrm {domain} (f)}$ and ${\displaystyle a\in I}$, then $${\displaystyle {\begin{aligned}\lim _{x\to a^{+}}f(x)=R&\iff \forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,0<x-a<\delta \longrightarrow |f(x)-R|<\varepsilon ,\\\lim _{x\to a^{-}}f(x)=L&\iff \forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,0<a-x<\delta \longrightarrow |f(x)-L|<\varepsilon .\end{aligned}}}$$

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.

Example 1. The limits from the left and from the right of ${\textstyle g(x):=-{\frac {1}{x}}}$ as ${\displaystyle x}$ approaches ${\displaystyle a:=0}$ are, respectively $${\displaystyle \lim _{x\to 0^{-}}-{\frac {1}{x}}=+\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}{-1/x}=-\infty .}$$ The reason why ${\textstyle \lim _{x\to 0^{-}}-{\frac {1}{x}}=+\infty }$ is because ${\displaystyle x}$ is always negative (since ${\displaystyle x\to 0^{-}}$ means that ${\displaystyle x\to 0}$ with all values of ${\displaystyle x}$ satisfying ${\displaystyle x<0}$), which implies that ${\displaystyle -1/x}$ is always positive so that ${\textstyle \lim _{x\to 0^{-}}-{\frac {1}{x}}}$ diverges^{note 1} to ${\displaystyle +\infty }$ (and not to ${\displaystyle -\infty }$) as ${\displaystyle x}$ approaches ${\displaystyle 0}$ from the left. Similarly, ${\textstyle \lim _{x\to 0^{+}}-{\frac {1}{x}}=-\infty }$ since all values of ${\displaystyle x}$ satisfy ${\displaystyle x>0}$ (said differently, ${\displaystyle x}$ is always positive) as ${\displaystyle x}$ approaches ${\displaystyle 0}$ from the right, which implies that ${\displaystyle -1/x}$ is always negative so that ${\textstyle \lim _{x\to 0^{+}}-{\frac {1}{x}}}$ diverges to ${\displaystyle -\infty .}$

Example 2. One example of a function with different one-sided limits is ${\textstyle f(x)={\frac {1}{1+2^{-1/x}}}}$, 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 $${\displaystyle \lim _{x\to 0^{-}}2^{-1/x}=\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}2^{-1/x}=0,}$$ which is true because ${\textstyle \lim _{x\to 0^{-}}{-1/x}=+\infty }$ and ${\textstyle \lim _{x\to 0^{+}}{-1/x}=-\infty }$ so that consequently, $${\displaystyle \lim _{x\to 0^{+}}{\frac {1}{1+2^{-1/x}}}={\frac {1}{1+\displaystyle \lim _{x\to 0^{+}}2^{-1/x}}}={\frac {1}{1+0}}=1}$$ whereas ${\textstyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0}$ because the denominator diverges to infinity; that is, because ${\displaystyle \lim _{x\to 0^{-}}1+2^{-1/x}=\infty }$. Since ${\displaystyle \lim _{x\to 0^{-}}f(x)\neq \lim _{x\to 0^{+}}f(x)}$, the limit ${\displaystyle \lim _{x\to 0}f(x)}$ does not exist.

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 Alternatively, one may consider the domain with a half-open interval topology.

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.