TRIAC

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 below or from above. One should write either:

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

for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly

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

for the limit as x increases in value approaching a (x approaches a "from the left" or "from below")

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

$\underset{x\to a}{\lim }f(x)$

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 left-sided limit can be rigorously defined as:

$\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0\mathrm{\forall }x\in I(0<a-x<\delta \Rightarrow |f(x)-L|<\epsilon )$

Similarly, the right-sided limit can be rigorously defined as:

$\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0\mathrm{\forall }x\in I(0<x-a<\delta \Rightarrow |f(x)-L|<\epsilon )$

Where $I$ represents some interval that is within the domain of $f$

Examples

One example of a function with different one-sided limits is the following:

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

whereas

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

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.

See also

• Real projective line

External links

• Template:Planetmath reference