This article needs additional citations for verification. (August 2021)
The limit as x decreases in value approaching a (x approaches a citation needed] or "from above") can be denoted:[
The limit as x increases in value approaching a (x approaches a citation needed] or "from below") can be denoted:[
citation needed] it is common to use the short notation:[
- for the left limit and for the right limit.
The two one-sided limits exist and are equal if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a "two-sided limit".
In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.
The right-sided limit can be rigorously defined as
and the left-sided limit can be rigorously defined as
One example of a function with different one-sided limits is the following (cf. picture):
Relation to topological definition of limitEdit
The one-sided limit to a point 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 p.[verification needed] Alternatively, one may consider the domain with a half-open interval topology.
- "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.
- 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.
- "one-sided limit". planetmath.org. 22 March 2013. Archived from the original on 26 January 2021. Retrieved 7 August 2021.
- 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.