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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 writes either## Examples

_{0} = 3 the limit from the left is
_{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

## See also

## External links

- $lim\_\{xto\; a^+\}f(x)\; mathrm\{or\}\; lim\_\{xdownarrow\; a\},f(x)$

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

- $lim\_\{xto\; a^-\}f(x)\; mathrm\{or\}\; lim\_\{xuparrow\; a\},\; f(x)$

for the limit as x approaches a from below (or "from the left").

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

- $lim\_\{xto\; a\}\; f(x),$

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.

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

- $lim\_\{x\; rarr\; 0^-\}\{1\; over\; 1\; +\; 2^\{-1/x\}\}\; =\; 0.$

Another example is the piecewise function

- $f(x)=left\{begin\{matrix\}x^2\; \&\; mbox\{\; for\; \}\; x<\; 3\; 11-(x-3)^2\&\; mbox\{\; for\; \}\; x>3end\{matrix\}right.$

- $lim\_\{xrarr\; 3^-\}\; f(x)\; =\; 9$

- $lim\_\{xrarr\; 3^+\}\; f(x)\; =\; 11.$

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.

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Last updated on Thursday October 09, 2008 at 14:16:49 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday October 09, 2008 at 14:16:49 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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