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| <!--[[WP:MSM]] forbids the use of mathematical definitions that use the notation of symbolic logic.-->
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| In [[calculus]], a '''one-sided limit''' is either of the two [[Limit of a function|limits]] of a [[function (mathematics)|function]] ''f''(''x'') of a [[real number|real]] variable ''x'' as ''x'' approaches a specified point either from below or from above. One should write either:
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| :<math>\lim_{x\to a^+}f(x)\ </math> or <math> \lim_{x\downarrow a}\,f(x)</math> or <math> \lim_{x \searrow a}\,f(x)</math>
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| for the limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' "from the right" or "from above"), and similarly
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| :<math>\lim_{x\to a^-}f(x)\ </math> or <math> \lim_{x\uparrow a}\, f(x)</math> or <math> \lim_{x \nearrow a}\,f(x)</math> | |
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| for the limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' "from the left" or "from below") | |
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| 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
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| :<math>\lim_{x\to a} f(x)\,</math>
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| 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.
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| The left-sided limit can be rigorously defined as:
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| :<math>\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon)</math>
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| Similarly, the right-sided limit can be rigorously defined as:
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| :<math>\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < x - a < \delta \Rightarrow |f(x) - L|<\varepsilon)</math> | |
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| Where <math> I </math> represents some interval that is within the domain of <math>f</math>
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| ==Examples==
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| One example of a function with different one-sided limits is the following:
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| :<math>\lim_{x \rarr 0^+}{1 \over 1 + 2^{-1/x}} = 1,</math> | |
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| whereas
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| :<math>\lim_{x \rarr 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math>
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| ==Relation to topological definition of limit==
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| The one-sided limit to a point ''p'' corresponds to the [[Limit_of_a_function#Functions_on_topological_spaces|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]].
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| ==Abel's theorem==
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| A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their [[radius of convergence|intervals of convergence]] is [[Abel's theorem]].
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| ==See also==
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| * [[Real projective line]]
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| ==External links==
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| *{{planetmath reference|title=One-sided limit|id=2950}}
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| [[Category:Real analysis]]
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| [[Category:Limits (mathematics)]]
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| [[Category:Functions and mappings]]
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As a leading authority on international business developments including ?big statistics?, entrepreneurs and the social media movement, I try to provide only the optimum quality and most sound information. I'm the creator of the award winning publication, How To Internet Market for the Self-Employed and a frequent speaker for many international specialists. I have spoken about international mega trends, big data and the social media revolution at seminars and business gatherings across the world.
my weblog :: costa rica travel tours