One-sided limit

The function $f(x)=x^{2}+\operatorname {sign} (x),$ where $\operatorname {sign} (x)$ denotes the sign function, has a left limit of $-1,$ a right limit of $+1,$ and a function value of $0$ at the point $x=0.$

In calculus, a one-sided limit refers to either one 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.^[1]^[2]

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

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

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

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

If the limit of $f(x)$ as $x$ approaches $a$ exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit $\lim _{x\to a}f(x)$ does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as $x$ approaches $a$ is sometimes called a "two-sided limit".^{[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.

Formal definition

Definition

If $I$ represents some interval that is contained in the domain of $f$ and if $a$ is a point in $I$ then the right-sided limit as $x$ approaches $a$ can be rigorously defined as the value $R$ that satisfies:^[6]^{[verification needed]} ${\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<x-a<\delta {\text{ then }}|f(x)-R|<\varepsilon ,$ and the left-sided limit as $x$ approaches $a$ can be rigorously defined as the value $L$ that satisfies: ${\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<a-x<\delta {\text{ then }}|f(x)-L|<\varepsilon .$

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

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

\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 ))

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:

\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 $x$ and $a$ is

$|x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|.$

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

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

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

Examples

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

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

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 $p.$ ^[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]}

Notes

^ A limit that is equal to $\infty$ is said to diverge to $\infty$ rather than converge to $\infty .$ The same is true when a limit is equal to $-\infty .$

References

^ ^a ^b ^c ^d "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.
^ ^a ^b ^c Fridy, J. A. (24 January 2020). Introductory Analysis: The Theory of Calculus. Gulf Professional Publishing. p. 48. ISBN 978-0-12-267655-0. Retrieved 7 August 2021.
^ Hasan, Osman; Khayam, Syed (2014-01-02). "Towards Formal Linear Cryptanalysis using HOL4" (PDF). Journal of Universal Computer Science. 20 (2): 209. doi:10.3217/jucs-020-02-0193. ISSN 0948-6968.
^ Gasic, Andrei G. (2020-12-12). Phase Phenomena of Proteins in Living Matter (Thesis thesis).
^ Brokate, Martin; Manchanda, Pammy; Siddiqi, Abul Hasan (2019), "Limit and Continuity", Calculus for Scientists and Engineers, Industrial and Applied Mathematics, Singapore: Springer Singapore, pp. 39–53, doi:10.1007/978-981-13-8464-6_2, ISBN 978-981-13-8463-9, S2CID 201484118, retrieved 2022-01-11
^ Giv, Hossein Hosseini (28 September 2016). Mathematical Analysis and Its Inherent Nature. American Mathematical Soc. p. 130. ISBN 978-1-4704-2807-5. Retrieved 7 August 2021.