One-sided limit

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:

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

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

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

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

\( \lim_{x\to a} f(x)\, \)

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:

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

Similarly, 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

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

\( \lim_{x \to 0^+}{1 \over 1 + 2^{-1/x}} = 1, \)

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.
See also

Real projective line
Semi-differentiability
Limit superior and limit inferior

External links

One-sided limit at PlanetMath.org.

Mathematics Encyclopedia