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One-sided limit

Summary

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 ${\displaystyle x=0.}$

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

Formal 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
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

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

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]

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. ^ a b c d "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.{{cite web}}: CS1 maint: url-status (link)
2. ^ 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.
3. ^ Hasan, Osman; Khayam, Syed (2014-01-02). "Towards Formal Linear Cryptanalysis using HOL4" (PDF). JUCS - Journal of Universal Computer Science. 20(2): 209. doi:10.3217/jucs-020-02-0193. ISSN 0948-6968.
4. ^ a b c d "one-sided limit". planetmath.org. 22 March 2013. Archived 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, 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.