Real Function Limits:One-Sided

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.

The limit as x decreases in value approaching a (x approaches a from the right or "from above") can be denoted:

$\lim _{x\to a^{+}}f(x)\$ or $\lim _{x\,\downarrow \,a}\,f(x)$ or $\lim _{x\searrow a}\,f(x)$ or $\lim _{x{\underset {>}{\to }}a}f(x)$

The limit as x increases in value approaching a (x approaches a "from the left" or "from below") can be denoted:

$\lim _{x\to a^{-}}f(x)\$ or $\lim _{x\,\uparrow \,a}\,f(x)$ or $\lim _{x\nearrow a}\,f(x)$ or $\lim _{x{\underset {<}{\to }}a}f(x)$

In probability theory it is common to use the short notation:

$f(x-)$ for the left limit and $f(x+)$ 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

$\lim _{x\to a}f(x)\,$

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

$\forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<x-a<\delta \Rightarrow |f(x)-L|<\varepsilon ),$

and the left-sided limit can be rigorously defined as

$\forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;(0<a-x<\delta \Rightarrow |f(x)-L|<\varepsilon ),$

where I represents some interval that is within the domain of f.

Examples

Plot of the function $1/(1+2^{-1/x})$

One example of a function with different one-sided limits is the following (cf. picture):

$\lim _{x\to 0^{+}}{1 \over 1+2^{-1/x}}=1,$

whereas

$\lim _{x\to 0^{-}}{1 \over 1+2^{-1/x}}=0.$

Relation to topological definition of limit

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. Alternatively, one may consider the domain with a half-open interval topology.

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.

Reference

1. "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.
2. 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. "one-sided limit". planetmath.org. 22 March 2013. Archived from the original on 26 January 2021. Retrieved 7 August 2021.
4. 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.