One-sided limit

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The function f(x) = x² + sign(x) has a left limit of -1, a right limit of +1, and a function value of 0 at the point x = 0.

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.^[1][2]

The limit as x decreases in value approaching a (x approaches a "from the right"^{[citation needed]} or "from above") can be denoted:

${\displaystyle \lim _{x\to a^{+}}f(x)\ }$ or ${\displaystyle \lim _{x\,\downarrow \,a}\,f(x)}$ or ${\displaystyle \lim _{x\searrow a}\,f(x)}$ or ${\displaystyle \lim _{x{\underset {>}{\to }}a}f(x)}$^{[1][2][3][additional citation(s) needed]}

The limit as x increases in value approaching a (x approaches a "from the left"^{[citation needed]} or "from below") can be denoted:

${\displaystyle \lim _{x\to a^{-}}f(x)\ }$ or ${\displaystyle \lim _{x\,\uparrow \,a}\,f(x)}$ or ${\displaystyle \lim _{x\nearrow a}\,f(x)}$ or ${\displaystyle \lim _{x{\underset {<}{\to }}a}f(x)}$^{[1][2][3][additional citation(s) needed]}

In probability theory^{[citation needed]} it is common to use the short notation:

${\displaystyle f(x-)}$ for the left limit and ${\displaystyle f(x+)}$ for the right limit.^[3]

The two one-sided limits exist and are equal if the limit of f(x) as x approaches a exists.^[3] In some cases in which the limit

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

does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a "two-sided limit".^{[citation needed]}

In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.^{[citation needed]}

The right-sided limit can be rigorously defined as

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

and the left-sided limit can be rigorously defined as

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

where I represents some interval that is within the domain of f.^{[3][4][verification needed]}

Examples[]

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

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

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

whereas

${\displaystyle \lim _{x\to 0^{-}}{1 \over 1+2^{-1/x}}=0.}$^{[citation needed]}

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.^{[1][verification needed]} Alternatively, one may consider the domain with a half-open interval topology.^{[citation needed]}

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]}

References[]

See also[]

• Projectively extended real line
• Semi-differentiability
• Limit superior and limit inferior