Real Function Limits:One-Sided - Revision history

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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 the left or from the right.<ref name=":0">{{Cite web|title=One-sided limit - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/One-sided_limit|url-status=live|access-date=7 August 2021|website=encyclopediaofmath.org}}</ref><ref name=":1">{{Cite book|last=Fridy|first=J. A.|url=https://books.google.com/books?id=SaZYs-OKqJcC&newbks=0&printsec=frontcover&pg=PA48&dq=%22one-sided+limit%22&hl=en|title=Introductory Analysis: The Theory of Calculus|date=24 January 2020|publisher=Gulf Professional Publishing|isbn=978-0-12-267655-0|pages=48|language=en|access-date=7 August 2021}}</ref>

The limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' {{Citation needed span|text="from the right"|date=August 2021}} or "from above") can be denoted:

:<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> or <math>\lim_{x \underset{>}{\to} a}f(x)</math><ref name=":0" /><ref name=":1" /><ref name=":2">{{Cite web|date=22 March 2013|title=one-sided limit|url=https://planetmath.org/onesidedlimit|url-status=live|archive-url=https://web.archive.org/web/20210126131057/https://planetmath.org/onesidedlimit|archive-date=26 January 2021|access-date=7 August 2021|website=planetmath.org}}</ref>{{Additional citation needed|date=August 2021}}

The limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' {{Citation needed span|text="from the left"|date=August 2021}} or "from below") can be denoted:

:<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> or <math>\lim_{x \underset{<}{\to} a}f(x)</math><ref name=":0" /><ref name=":1" /><ref name=":2" />{{Additional citation needed|date=August 2021}}

{{Citation needed span|text=In probability theory|date=August 2021}} it is common to use the short notation:

:<math>f(x-)</math> for the left limit and <math>f(x+)</math> for the right limit.<ref name=":2" />

The two one-sided limits exist and are equal if the limit of ''f''(''x'') as ''x'' approaches ''a'' exists.<ref name=":2" /> In some cases in which the limit

:<math>\lim_{x\to a} f(x)\,</math>

does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as ''x'' approaches ''a'' is sometimes called a "two-sided limit".{{Citation needed|date=August 2021}}

In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.{{Citation needed|date=August 2021}}

The right-sided limit can be rigorously defined as

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

and the left-sided limit can be rigorously defined as

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

where {{mvar|I}} represents some [[interval (mathematics)|interval]] that is within the [[domain of a function|domain]] of {{mvar|f}}.<ref name=":2" /><ref>{{Cite book|last=Giv|first=Hossein Hosseini|url=https://books.google.com/books?id=Hf0mDQAAQBAJ&newbks=0&printsec=frontcover&dq=%22one-sided+limit%22&hl=en|title=Mathematical Analysis and Its Inherent Nature|date=28 September 2016|publisher=American Mathematical Soc.|isbn=978-1-4704-2807-5|pages=130|language=en|access-date=7 August 2021}}</ref>{{Verify source|date=August 2021}}

==Examples==

[[File:1 div (1 + 2 ** (-1 div x)).svg|thumb|350px|Plot of the function <math>1 / (1 + 2^{-1/x})</math>]]
One example of a function with different one-sided limits is the following (cf. picture):

:<math>\lim_{x \to 0^+}{1 \over 1 + 2^{-1/x}} = 1,</math>

whereas

:<math>\lim_{x \to 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math>{{Citation needed|date=August 2021}}

==Relation to topological definition of limit==
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''.<ref name=":0" />{{Verify source|date=August 2021}} Alternatively, one may consider the domain with a [[half-open interval topology]].{{Citation needed|date=August 2021}}

==Abel's theorem==
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]].{{Citation needed|date=August 2021}}

==Resources==
* [https://en.wikipedia.org/wiki/One-sided_limit One-sided limit], Wikipedia

@@ Line 48: / Line 48: @@
 A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.
-==Reference==
+== Licensing ==
-# "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.
+Content obtained and/or adapted from:
-# Fridy, J. A. (24 January 2020). Introductory Analysis: The Theory of Calculus. Gulf Professional Publishing. p. 48. ISBN 978-0-12-267655-0. Retrieved 7 August 2021.
+* [https://en.wikipedia.org/wiki/One-sided_limit One-sided limit, Wikipedia] under a CC BY-SA license

@@ Line 1: / Line 1: @@
-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 the left or from the right.<ref name=":0">{{Cite web|title=One-sided limit - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/One-sided_limit|url-status=live|access-date=7 August 2021|website=encyclopediaofmath.org}}</ref><ref name=":1">{{Cite book|last=Fridy|first=J. A.|url=https://books.google.com/books?id=SaZYs-OKqJcC&newbks=0&printsec=frontcover&pg=PA48&dq=%22one-sided+limit%22&hl=en|title=Introductory Analysis: The Theory of Calculus|date=24 January 2020|publisher=Gulf Professional Publishing|isbn=978-0-12-267655-0|pages=48|language=en|access-date=7 August 2021}}</ref>
+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 the left or from the right.
-The limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' {{Citation needed span|text="from the right"|date=August 2021}} or "from above") can be denoted:
+The limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' ''from the right'' or "from above") can be denoted:
-:<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> or <math>\lim_{x \underset{>}{\to} a}f(x)</math><ref name=":0" /><ref name=":1" /><ref name=":2">{{Cite web|date=22 March 2013|title=one-sided limit|url=https://planetmath.org/onesidedlimit|url-status=live|archive-url=https://web.archive.org/web/20210126131057/https://planetmath.org/onesidedlimit|archive-date=26 January 2021|access-date=7 August 2021|website=planetmath.org}}</ref>{{Additional citation needed|date=August 2021}}
+:<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> or <math>\lim_{x \underset{>}{\to} a}f(x)</math>
-The limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' {{Citation needed span|text="from the left"|date=August 2021}} or "from below") can be denoted:
+The limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' "from the left" or "from below") can be denoted:
-:<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> or <math>\lim_{x \underset{<}{\to} a}f(x)</math><ref name=":0" /><ref name=":1" /><ref name=":2" />{{Additional citation needed|date=August 2021}}
+:<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> or <math>\lim_{x \underset{<}{\to} a}f(x)</math>
-{{Citation needed span|text=In probability theory|date=August 2021}} it is common to use the short notation:
+In probability theory it is common to use the short notation:
-:<math>f(x-)</math> for the left limit and <math>f(x+)</math> for the right limit.<ref name=":2" />
+:<math>f(x-)</math> for the left limit and <math>f(x+)</math> for the right limit.
-The two one-sided limits exist and are equal if the limit of ''f''(''x'') as ''x'' approaches ''a'' exists.<ref name=":2" />  In some cases in which the limit
+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
 :<math>\lim_{x\to a} f(x)\,</math>
-does not exist, the two one-sided limits nonetheless exist.  Consequently, the limit as ''x'' approaches ''a'' is sometimes called a "two-sided limit".{{Citation needed|date=August 2021}}
+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.{{Citation needed|date=August 2021}}
+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
@@ Line 29: / Line 29: @@
 :<math>\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon),</math>
-where {{mvar|I}} represents some [[interval (mathematics)|interval]] that is within the [[domain of a function|domain]] of {{mvar|f}}.<ref name=":2" /><ref>{{Cite book|last=Giv|first=Hossein Hosseini|url=https://books.google.com/books?id=Hf0mDQAAQBAJ&newbks=0&printsec=frontcover&dq=%22one-sided+limit%22&hl=en|title=Mathematical Analysis and Its Inherent Nature|date=28 September 2016|publisher=American Mathematical Soc.|isbn=978-1-4704-2807-5|pages=130|language=en|access-date=7 August 2021}}</ref>{{Verify source|date=August 2021}}
+where {{mvar|I}} represents some interval that is within the domain of {{mvar|f}}.
 ==Examples==
@@ Line 40: / Line 40: @@
 whereas
-:<math>\lim_{x \to 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math>{{Citation needed|date=August 2021}}
+:<math>\lim_{x \to 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math>
 ==Relation to topological definition of limit==
-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''.<ref name=":0" />{{Verify source|date=August 2021}} Alternatively, one may consider the domain with a [[half-open interval topology]].{{Citation needed|date=August 2021}}
+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 [[radius of convergence|intervals of convergence]] is [[Abel's theorem]].{{Citation needed|date=August 2021}}
+A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.
-==Resources==
+==Reference==
-* [https://en.wikipedia.org/wiki/One-sided_limit One-sided limit], Wikipedia
+# "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 7 August 2021.