View source for Limit (mathematics)

{{dablink|This is a partial overview of the idea of a limit in mathematics. For still more specific uses of "limit", see [[Limit of a sequence]] and [[Limit of a function]].}}

In [[mathematics]], a '''limit''' is the value that a [[function (mathematics)|function]] (or [[sequence]]) "approaches" as the input (or index) "approaches" some [[Value (mathematics)|value]].<ref>{{cite book |last=Stewart |first=James |authorlink=James Stewart (mathematician) |year=2008 |title=Calculus: Early Transcendentals |edition=6th |publisher=[[Brooks/Cole]] |isbn=978-0-495-01166-8 |url-access=registration |url=https://archive.org/details/calculusearlytra00stew_1 }}</ref> Limits are essential to [[calculus]] (and [[mathematical analysis]] in general) and are used to define [[continuous function|continuity]], [[derivative]]s, and [[integral]]s.

The concept of a [[limit of a sequence]] is further generalized to the concept of a limit of a [[net (topology)|topological net]], and is closely related to [[limit (category theory)|limit]] and [[direct limit]] in [[category theory]].

In formulas, a limit of a function is usually written as
:<math> \lim_{x \to c} f(x) = L,</math>
and is read as "the limit of {{math|''f''}} of {{math|''x''}} as {{mvar|''x''}} approaches {{math|''c''}} equals {{math|''L''}}". The fact that a function {{math|''f''}} approaches the limit {{math|''L''}} as {{math|''x''}} approaches {{math|''c''}} is sometimes denoted by a right arrow (→), as in
:<math>f(x) \to L \text{ as } x \to c.</math>

== Limit of a function ==
{{main|Limit of a function}}
[[File:Límite 01.svg|thumb|Whenever a point {{mvar|x}} is within a distance {{mvar|δ}} of {{mvar|c}}, the value {{math|''f''(''x'')}} is within a distance {{mvar|ε}} of {{mvar|L}}.]]
[[File:Limit-at-infinity-graph.png|thumb|For all {{math|''x'' &gt; ''S''}}, the value {{math|''f''(''x'')}} is within a distance {{mvar|ε}} of {{math|''L''}}.]]

Suppose {{math|''f''}} is a [[real-valued function]] and {{math|''c''}} is a [[real number]]. Intuitively speaking, the expression

:<math> \lim_{x \to c}f(x) = L </math>

means that {{math|''f''(''x'')}} can be made to be as close to {{math|''L''}} as desired by making {{math|''x''}} sufficiently close to {{math|''c''}}.  In that case, the above equation can be read as "the limit of {{math|''f''}} of {{math|''x''}}, as {{math|''x''}} approaches {{math|''c''}}, is {{math|''L''}}". 

[[Augustin-Louis Cauchy]] in 1821,<ref name=Larson>{{cite book |last1=Larson |first1=Ron |authorlink1=Ron Larson (mathematician) |last2=Edwards |first2=Bruce H. |year=2010 |title=Calculus of a single variable |edition=Ninth |publisher=[[Brooks/Cole]], [[Cengage Learning]] |isbn=978-0-547-20998-2}}</ref> followed by [[Karl Weierstrass]], formalized the definition of the limit of a function which became known as the [[(ε, δ)-definition of limit]]. The definition uses {{math|[[ε]]}} (the lowercase Greek letter ''epsilon'') to represent any small positive number, so that "{{math|''f''(''x'')}} becomes arbitrarily close to {{math|''L''}}" means that {{math|''f''(''x'')}} eventually lies in the interval {{math|(''L'' − ε, ''L'' + ε)}}, which can also be written using the absolute value sign as {{math|{{abs|''f''(''x'') − ''L''}} < ε}}.<ref name=Larson/> The phrase "as {{math|''x''}} approaches {{math|''c''}}" then indicates that we refer to values of {{math|''x''}} whose distance from {{math|''c''}} is less than some positive number {{math|[[δ]]}} (the lower case Greek letter ''delta'')—that is, values of {{math|''x''}} within either {{math|(''c'' − δ, ''c'')}} or {{math|(''c'', ''c'' + δ)}}, which can be expressed with {{math|0 < {{abs|''x'' − ''c''}} < δ}}. The first inequality means that the distance between {{math|''x''}} and {{math|''c''}} is greater than {{math|0}} and that {{math|''x'' ≠ ''c''}}, while the second indicates that {{math|''x''}} is within distance {{math|δ}} of {{math|''c''}}.<ref name=Larson/>

The above definition of a limit is true even if {{math|''f''(''c'') ≠ ''L''}}. Indeed, the function {{math|''f''}} need not even be defined at {{math|''c''}}.

For example, if

:<math> f(x) = \frac{x^2 - 1}{x - 1} </math>

then {{math|''f''(1)}} is not defined (see [[indeterminate forms]]), yet as {{math|''x''}} moves arbitrarily close to 1, {{math|''f''(''x'')}} correspondingly approaches 2:

{| class="wikitable"
|''f''(0.9)||''f''(0.99)||''f''(0.999)|| ''f''(1.0) ||''f''(1.001)||''f''(1.01)||''f''(1.1)
|-
|    1.900 ||     1.990 ||      1.999 ||  undefined ||      2.001 ||     2.010 ||    2.100
|}

Thus, {{math|''f''(''x'')}} can be made arbitrarily close to the limit of 2 just by making {{math|''x''}} sufficiently close to {{math|1}}.

In other words, <math> \lim_{x \to 1} \frac{x^2-1}{x-1} = 2 </math>

This can also be calculated algebraically, as <math>\frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1</math> for all real numbers {{math|''x'' ≠ 1}}.

Now since {{math|''x'' + 1}} is continuous in {{math|''x''}} at 1, we can now plug in 1 for {{math|''x''}}, thus <math>\lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2</math>.

In addition to limits at finite values, functions can also have limits at infinity. For example, consider
:<math>f(x) = {2x-1 \over x}</math>
* ''f''(100) = 1.9900
* ''f''(1000) = 1.9990
* ''f''(10000) = 1.99990

As {{math|''x''}} becomes extremely large, the value of {{math|''f''(''x'')}} approaches 2, and the value of {{math|''f''(''x'')}} can be made as close to 2 as one could wish just by picking {{math|''x''}} sufficiently large. So in this case, the limit of {{math|''f''(''x'')}} as {{math|''x''}} approaches infinity is 2. In mathematical notation,

:<math> \lim_{x \to \infty} \frac{2x-1}{x} = 2. </math>

== Limit of a sequence ==
{{main|Limit of a sequence}}

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.

Formally, suppose {{math|''a''<sub>1</sub>, ''a''<sub>2</sub>, ...}} is a [[sequence]] of [[real number]]s. It can be stated that the real number {{math|''L''}} is the ''limit'' of this sequence, namely:

:<math> \lim_{n \to \infty} a_n = L </math>

which is read as
:"The limit of ''a<sub>n</sub>'' as ''n'' approaches infinity equals ''L''"

to mean

:For every [[real number]] {{math|ε > 0}}, there exists a [[natural number]] {{math|''N''}} such that for all {{math|''n'' > ''N''}}, we have {{math|{{abs|''a''<sub>''n''</sub> − {{math|''L''}}}} < ε}}.

Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the [[absolute value]] {{math|{{abs|''a''<sub>''n''</sub> − {{math|''L''}}}}}} is the distance between {{math|''a''<sub>''n''</sub>}} and {{math|''L''}}. Not every sequence has a limit; if it does, it is called ''[[Convergent series|convergent]]'', and if it does not, it is ''divergent''. One can show that a convergent sequence has only one limit.

The limit of a sequence and the limit of a function are closely related.  On the one hand, the limit as {{math|''n''}} approaches infinity of a sequence {{math|{{mset|''a''<sub>''n''</sub>}}}} is simply the limit at infinity of a function {{math|''a''(''n'')}} defined on the [[natural number]]s {{math|{{mset|''n''}}}}. On the other hand, if {{math|''X''}} is the domain of a function {{math|''f''(''x'')}} and if the limit as {{math|''n''}} approaches infinity of {{math|''f''(''x''<sub>''n''</sub>)}} is {{math|''L''}} for ''every'' arbitrary sequence of points {{math|{{mset|''x''<sub>''n''</sub>}}}} in {{math|{{mset|''X'' – {{mset|''x''<sub>0</sub>}}}}}} which converges to {{math|''x''<sub>0</sub>}}, then the limit of the function {{math|''f''(''x'')}} as {{math|''x''}} approaches {{math|''x''<sub>0</sub>}} is {{math|''L''}}.<ref>{{harvtxt|Apostol|1974|pp=75–76}}</ref> One such sequence would be {{math|{{mset|''x''<sub>0</sub> + 1/''n''}}}}.

==Limit as "standard part"==
In [[non-standard analysis]] (which involves a [[hyperreal number|hyperreal]] enlargement of the number system), the limit of a sequence <math>(a_n)</math> can be expressed as the [[standard part function|standard part]] of the value <math>a_H</math> of the natural extension of the sequence at an infinite [[hypernatural]] index ''n=H''.  Thus,
:<math> \lim_{n \to \infty} a_n = \operatorname{st}(a_H) </math>.
Here the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is [[infinitesimal]]).  This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence.  Conversely, the standard part of a hyperreal <math>a=[a_n]</math> represented in the ultrapower construction by a Cauchy sequence <math>(a_n)</math>, is simply the limit of that sequence:
:<math> \operatorname{st}(a)=\lim_{n \to \infty} a_n </math>.
In this sense, taking the limit and taking the standard part are equivalent procedures.

==Convergence and fixed point==
A formal definition of convergence can be stated as follows.
Suppose  <math> { {p}_{n} } </math> as <math> n </math> goes from <math> 0 </math> to <math>  \infty  </math> is a sequence that converges to <math> p </math>, with <math>  {p}_{n} \neq p </math> for all <math> n </math>. If positive constants <math> \lambda </math> and <math> \alpha </math> exist with
::::::<math>\lim_{n \rightarrow  \infty  }  \frac{ \left | { p}_{n+1 } -p   \right |  }{ { \left |  { p}_{n }-p   \right |  }^{ \alpha} } =\lambda </math>
then <math> { {p}_{n} } </math> as <math> n </math> goes from <math> 0 </math> to <math>  \infty  </math> converges to <math> p </math> of order <math> \alpha </math>, with asymptotic error constant <math> \lambda </math>

Given a function <math> f </math> with a fixed point <math> p </math>, there is a nice checklist for checking the convergence of the sequence <math>p_n</math>.

:1) First check that p is indeed a fixed point:
::<math> f(p) = p </math>
:2) Check for linear convergence.  Start by finding <math>\left | f^\prime (p)  \right | </math>.  If....
{| class="wikitable" border="1"

| <math>\left | f^\prime (p)  \right | \in (0,1)</math>
| then there is linear convergence
|-
| <math>\left | f^\prime (p)  \right | > 1</math>
| series diverges
|-
| <math>\left | f^\prime (p)  \right | =0 </math>
| then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence
|}
:3) If it is found that there is something better than linear the expression should be checked for quadratic convergence.  Start by finding <math>\left | f^{\prime\prime} (p)  \right | </math> If....
{| class="wikitable" border="1"
| <math>\left | f^{\prime\prime} (p)  \right | \neq 0</math>
| then there is quadratic convergence provided that <math> f^{\prime\prime} (p) </math>  is continuous
|-
| <math>\left | f^{\prime\prime} (p)  \right | = 0</math>
| then there is something even better than quadratic convergence
|-
| <math>\left | f^{\prime\prime} (p)  \right | </math> does not exist
| then there is convergence that is better than linear but still not quadratic
|}
<ref>''Numerical Analysis'', 8th Edition, Burden and Faires, Section 2.4 Error Analysis for Iterative Methods </ref>

== Computability of the limit ==
Limits can be difficult to compute. There exist limit expressions whose [[modulus of convergence]] is [[Undecidable problem|undecidable]]. In [[recursion theory]], the [[Limit lemma]] proves that it is possible to encode undecidable problems using limits.<ref>''Recursively enumerable sets and degrees'', Soare, Robert I.</ref>

== See also ==
{{wikibooks|Calculus|Limits}}
* [[Rate of convergence]]: the rate at which a convergent sequence approaches its limit
* [[Cauchy sequence]]
** [[Complete metric space]]
* [[Limit of a function]]
** [[One-sided limit]]: either of the two limits of functions of a real variable ''x'', as ''x'' approaches a point from above or below
** [[List of limits]]: list of limits for common functions
** [[Squeeze theorem]]: finds a limit of a function via comparison with two other functions
* [[Modes of convergence]]
** An [[Modes of convergence (annotated index)|annotated index]]
* [[Convergence of random variables]]
* [[Banach limit]] defined on the Banach space that extends the usual limits.
* [[Limit (category theory)|Limit in category theory]]
** [[Direct limit]]
** [[Inverse limit]]
* [[Asymptotic analysis]]: a method of describing limiting behavior
** [[Big O notation]]: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity
* [[Convergent matrix]]

==Notes==
{{reflist}}

== References ==
* {{ citation | last1 = Apostol | first1 = Tom M. | year = 1974 | lccn = 72011473 | title = Mathematical Analysis | edition = 2nd | publisher = [[Addison-Wesley]] | location = Menlo Park }}

== External links ==
{{Library resources box
|by=no
|onlinebooks=no
|others=no
|about=yes
|label=Limit (mathematics)}}
* {{MathWorld |title=Limit |urlname=Limit}}
* [http://www.mathwords.com/l/limit.htm Mathwords: Limit]

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[[Category:Limits (mathematics)| ]]
[[Category:Convergence (mathematics)]]
[[Category:Real analysis]]
[[Category:Asymptotic analysis]]
[[Category:Differential calculus]]
[[Category:General topology]]