One-sided limit

Summary

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

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

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Definition

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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

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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

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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

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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

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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

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  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

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  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

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