# One-sided limit

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In calculus, a **one-sided limit** refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right.^{[1]}^{[2]}

The limit as decreases in value approaching ( approaches "from the right"^{[3]} or "from above") can be denoted:^{[1]}^{[2]}

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

If the limit of as approaches exists then the limits from the left and from the right both exist and are equal. In some cases in which the 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[edit]

### Definition[edit]

If represents some interval that is contained in the domain of and if is a point in then the right-sided limit as approaches can be rigorously defined as the value that satisfies:^{[6]}^{[verification needed]}

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

Let represent an interval, where , and .

### Intuition[edit]

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:

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between and is

For the limit from the right, we want to be to the right of , which means that , so is positive. From above, is the distance between and . We want to bound this distance by our value of , giving the inequality . Putting together the inequalities and and using the transitivity property of inequalities, we have the compound inequality .

Similarly, for the limit from the left, we want to be to the left of , which means that . In this case, it is that is positive and represents the distance between and . Again, we want to bound this distance by our value of , leading to the compound inequality .

Now, when our value of is in its desired interval, we expect that the value of is also within its desired interval. The distance between and , the limiting value of the left sided limit, is . Similarly, the distance between and , the limiting value of the right sided limit, is . In both cases, we want to bound this distance by , so we get the following: for the left sided limit, and for the right sided limit.

## Examples[edit]

* Example 1*:
The limits from the left and from the right of as approaches are

^{[note 1]}to (and not to ) as approaches from the left. Similarly, since all values of satisfy (said differently, is always positive) as approaches from the right, which implies that is always negative so that diverges to

* Example 2*:
One example of a function with different one-sided limits is (cf. picture) where the limit from the left is and the limit from the right is
To calculate these limits, first show that

## Relation to topological definition of limit[edit]

The one-sided limit to a point 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 ^{[1]}^{[verification needed]} Alternatively, one may consider the domain with a half-open interval topology.^{[citation needed]}

## Abel's theorem[edit]

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[edit]

**^**A limit that is equal to is said to*di*verge to rather than*con*verge to The same is true when a limit is equal to

## References[edit]

- ^
^{a}^{b}^{c}^{d}"One-sided limit - Encyclopedia of Mathematics".*encyclopediaofmath.org*. Retrieved 7 August 2021.`{{cite web}}`

: CS1 maint: url-status (link) - ^
^{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. **^**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.**^**Gasic, Andrei G. (2020-12-12).*Phase Phenomena of Proteins in Living Matter*(Thesis thesis).**^**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**^**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.