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.^[1][2]

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

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][4]

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

The limit as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left"^[5][6] or "from below") can be denoted:^[1][2][4]

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

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.^[4] 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.

Formal definition

Definition

If ${\displaystyle I}$  represents some interval that is contained in the domain of ${\displaystyle f}$  and if ${\displaystyle a}$  is 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:^{[4][7][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:

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

 

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

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

 

The reason why ${\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty }$  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 ${\displaystyle \lim _{x\to 0^{-}}{-1/x}}$  diverges^{[note 1]} to ${\displaystyle +\infty }$  (and not to ${\displaystyle -\infty }$ ) as ${\displaystyle x}$  approaches ${\displaystyle 0}$  from the left. Similarly, ${\displaystyle \lim _{x\to 0^{+}}{-1/x}=-\infty }$  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 ${\displaystyle \lim _{x\to 0^{+}}{-1/x}}$  diverges to ${\displaystyle -\infty .}$ 

 

Plot of the function ${\displaystyle 1/(1+2^{-1/x}).}$ 

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

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

 

(which is true because ${\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty {\text{ and }}\lim _{x\to 0^{+}}{-1/x}=-\infty }$ ) 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 ${\displaystyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0}$  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.

Relation to topological definition of limit

See also: Filters in topology

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

Main article: 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

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

References

1. "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.{{cite web}}: CS1 maint: url-status (link)
2. 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.
3. 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.
4. "one-sided limit". planetmath.org. 22 March 2013. from the original on 26 January 2021. Retrieved 7 August 2021.
5. Gasic, Andrei G. (2020-12-12). Phase Phenomena of Proteins in Living Matter (Thesis thesis).
6. Brokate, Martin; Manchanda, Pammy; Siddiqi, Abul Hasan (2019), "Limit and Continuity", Calculus for Scientists and Engineers, 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
7. 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.

See also

• Projectively extended real line
• Semi-differentiability
• Limit superior and limit inferior