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Summary 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.

The limit as $x$ decreases in value approaching $a$ ($x$ approaches $a$ "from the right" or "from above") can be denoted:

$\lim _{x\to a^{+}}f(x)\quad {\text{ or }}\quad \lim _{x\,\downarrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\searrow a}\,f(x)\quad {\text{ or }}\quad f(x+)$ The limit as $x$ increases in value approaching $a$ ($x$ approaches $a$ "from the left" or "from below") can be denoted:

$\lim _{x\to a^{-}}f(x)\quad {\text{ or }}\quad \lim _{x\,\uparrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\nearrow a}\,f(x)\quad {\text{ or }}\quad f(x-)$ 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 exists (while the other does not exist). It is also possible for neither of the two one-sided limits to exists.

Formal definition

If $I$ represents some interval that is contained in the domain of $f$ and if $a$ is point in $I$ then the right-sided limit as $x$ approaches $a$ can be rigorously defined as the value $R$ that satisfies:[verification needed]

${\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 $x$ 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 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 $-\infty .$  Plot of the function $1/(1+2^{-1/x}).$ 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 $p.$ [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 $\infty$ is said to diverge to $\infty$ rather than converge to $\infty .$ The same is true when a limit is equal to $-\infty .$ 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.