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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)\ \mathrm{or}\ \lim_{x\downarrow 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)\ \mathrm{or}\ \lim_{x\uparrow 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 and only 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.

Examples

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

:

\lim_{x \rarr 0^+}{1 \over 1 + 2^{-1/x}} = 1,

whereas

:

\lim_{x \rarr 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''.

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.

one-sided_limit
Source: Wikipedia