One-sided limit

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]}

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

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

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

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. 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 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] [ 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.
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. Gasic, Andrei G. (2020-12-12). Phase Phenomena of Proteins in Living Matter (Thesis thesis).
5. 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
6. 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