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The function where denotes the sign function, has a left limit of a right limit of and a function value of at the point

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

does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as approaches 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 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]

and the left-sided limit as approaches can be rigorously defined as the value that satisfies:

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

Let represent an interval, where , and .

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:

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

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

The reason why is because is always negative (since means that with all values of satisfying ), which implies that is always positive so that diverges[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

Plot of the function

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

(which is true because ) so that consequently,
whereas because the denominator diverges to infinity; that is, because Since the limit does not exist.

Relation to topological definition of limit

See also: Filters in topology

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

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 is said to diverge to rather than converge to The same is true when a limit is equal to

References

  1. ^ a b c d "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.
  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). 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