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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 / (1 + 2^{-1/x})}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 0^+}{1 \over 1 + 2^{-1/x}} = 1,}

whereas

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Licensing

Content obtained and/or adapted from:

• One-sided limit, Wikipedia under a CC BY-SA license