# One-sided limit

Short description: Limit of a function approaching a value point from values below or above the value point
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] $\displaystyle{ \text{for all } \varepsilon \gt 0\;\text{ there exists some } \delta \gt 0 \;\text{ such that for all } x \in I, \text{ if } \;0 \lt x - a \lt \delta \text{ then } |f(x) - R| \lt \varepsilon, }$ and the left-sided limit as $\displaystyle{ x }$ approaches $\displaystyle{ a }$ can be rigorously defined as the value $\displaystyle{ L }$ that satisfies: $\displaystyle{ \text{for all } \varepsilon \gt 0\;\text{ there exists some } \delta \gt 0 \;\text{ such that for all } x \in I, \text{ if } \;0 \lt a - x \lt \delta \text{ then } |f(x) - L| \lt \varepsilon. }$

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

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

$\displaystyle{ \lim_{x \to a^{+}} f(x) = R ~~~ \iff ~~~ (\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, (0 \lt x - a \lt \delta \longrightarrow | f(x) - R | \lt \varepsilon)) }$
$\displaystyle{ \lim_{x \to a^{-}} f(x) = L ~~~ \iff ~~~ (\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, (0 \lt a - x \lt \delta \longrightarrow | f(x) - L | \lt \varepsilon)) }$

### 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:

$\displaystyle{ \lim_{x \to a} f(x) = L ~~~ \iff ~~~ \forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, 0 \lt |x - a| \lt \delta \implies | f(x) - L | \lt \varepsilon }$

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 $\displaystyle{ a \lt x }$, 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 $\displaystyle{ x - a \lt \delta }$. Putting together the inequalities $\displaystyle{ 0 \lt x - a }$ and $\displaystyle{ x - a \lt \delta }$ and using the transitivity property of inequalities, we have the compound inequality $\displaystyle{ 0 \lt x - a \lt \delta }$.

Similarly, for the limit from the left, we want $\displaystyle{ x }$ to be to the left of $\displaystyle{ a }$, which means that $\displaystyle{ x \lt a }$. 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 $\displaystyle{ 0 \lt a - x \lt \delta }$.

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: $\displaystyle{ |f(x) - L| \lt \varepsilon }$ for the left sided limit, and $\displaystyle{ |f(x) - R| \lt \varepsilon }$ 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 $\displaystyle{ x \lt 0 }$), 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 $\displaystyle{ x \gt 0 }$ (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. Fridy, J. A. (24 January 2020) (in en). Introductory Analysis: The Theory of Calculus. Gulf Professional Publishing. pp. 48. ISBN 978-0-12-267655-0. Retrieved 7 August 2021.
2. Hasan, Osman; Khayam, Syed (2014-01-02). "Towards Formal Linear Cryptanalysis using HOL4" (in en). Journal of Universal Computer Science 20 (2): 209. doi:10.3217/jucs-020-02-0193. ISSN 0948-6968.
3. Gasic, Andrei G. (2020-12-12). Phase Phenomena of Proteins in Living Matter (Thesis thesis).
4. Brokate, Martin; Manchanda, Pammy; Siddiqi, Abul Hasan (2019), "Limit and Continuity" (in en), 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, retrieved 2022-01-11
5. Giv, Hossein Hosseini (28 September 2016) (in en). Mathematical Analysis and Its Inherent Nature. American Mathematical Soc.. pp. 130. ISBN 978-1-4704-2807-5. Retrieved 7 August 2021.
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