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| | The family that wrote write-up is called Mireille and she or he totally digs that determine. I work as a librarian. Watching movies is one of many things I love most. For a while she's been in Puerto Rico. He is running and maintaining a blog here: http://usmerch.co.uk/mens-clothing/<br><br>my page: [http://usmerch.co.uk/mens-clothing/ gas monkey garage] |
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| {| class="wikitable" style="width:100%;"
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| !''n''!!''n'' sin(1/''n'')
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| |1||0.841471
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| |2||0.958851
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| |colspan="2"|...
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| |10||0.998334
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| |colspan="2"|...
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| |100||0.999983
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| |}
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| As the positive [[integer]] ''n'' becomes larger and larger, the value ''n'' sin(1/''n'') becomes arbitrarily close to 1. We say that "the limit of the sequence ''n'' [[sine|sin]](1/''n'') equals 1."
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| In [[mathematics]], the '''limit of a sequence''' is the value that the terms of a [[sequence]] "tend to".<ref name="Courant (1961), p. 29">Courant (1961), p. 29.</ref> If such a limit exists, the sequence is called '''convergent'''. A sequence which does not converge is said to be '''divergent'''.<ref>Courant (1961), p. 39.</ref> The limit of a sequence is said to be the fundamental notion on which the whole of [[Mathematical Analysis|analysis]] ultimately rests.<ref name="Courant (1961), p. 29"/>
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| Limits can be defined in any [[metric space|metric]] or [[topological space]], but are usually first encountered in the [[real number]]s.
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| ==Real numbers==
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| [[File:Converging Sequence example.svg|320px|thumb|The plot of a convergent sequence {''a<sub>n</sub>''} is shown in blue. Visually we can see that the sequence is converging to the limit 0 as ''n'' increases.]]
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| ===Formal Definition===
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| We call <math>x</math> the '''limit''' of the [[sequence]] <math>(x_n)</math> if the following condition holds:
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| :*For each [[real number]] <math>\epsilon > 0</math>, there exists a [[natural number]] <math>N</math> such that, for every natural number <math>n > N</math>, we have <math>|x_n - x| < \epsilon</math>.
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| In other words, for every measure of closeness <math>\epsilon</math>, the sequence's terms are eventually that close to the limit. The sequence <math>(x_n)</math> is said to '''converge to''' or '''tend to''' the limit <math>x</math>, written <math>x_n \to x</math> or <math>\lim_{n \to \infty} x_n = x</math>.
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| If a sequence converges to some limit, then it is '''convergent'''; otherwise it is '''divergent'''.
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| ===Examples===
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| *If <math>x_n = c</math> for some constant ''c'', then <math>x_n \to c</math>. ''Proof'': choose <math>N = 1</math>. We have that, for every <math>n > N</math>, <math>|x_n - c| = 0 < \epsilon</math>.
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| *If <math>x_n = 1/n</math>, then <math>x_n \to 0</math>. ''Proof'': choose <math>N = \left\lfloor\frac{1}{\epsilon}\right\rfloor</math> (the [[Floor and ceiling functions|floor function]]). We have that, for every <math>n > N</math>, <math>|x_n - 0| \le x_{N+1} = \frac{1}{\lfloor1/\epsilon\rfloor + 1} < \epsilon</math>.
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| *If <math>x_n = 1/n</math> when <math>n</math> is even, and <math>x_n = 1/n^2</math> when <math>n</math> is odd, then <math>x_n \to 0</math>. (The fact that <math>x_{n+1} > x_n</math> whenever <math>n</math> is odd is irrelevant.)
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| *Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence <math>0.3, 0.33, 0.333, 0.3333, ...</math> converges to <math>1/3</math>. Note that the [[decimal representation]] <math>0.3333...</math> is the ''limit'' of the previous sequence, defined by
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| :<math> 0.3333...\triangleq\lim_{n\to \infty} \sum_{i=1}^n \frac{3}{10^i}</math>.
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| *Finding <math>c</math> might sometimes be non-intuitive, like <math>\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>, the [[e (mathematical constant)|number ''e'']]. In these cases, one common approach is to find [[upper and lower bounds]] for the limit of the sequence (e.g., proving that <math>2.71< e <2.72</math>).
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| ===Properties===
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| Limits of sequences behave well with respect to the usual [[arithmetic operations]]. If <math>a_n \to a</math> and <math>b_n \to b</math>, then <math>a_n+b_n \to a+b</math>, <math>a_nb_n \to ab</math> and, if neither ''b'' nor any <math>b_n</math> is zero, <math>a_n/b_n \to a/b</math>.
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| For any [[continuous function]] ''f'', if <math>x_n \to x</math> then <math>f(x_n) \to f(x)</math>. In fact, any real-valued [[function (mathematics) | function]] ''f'' is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity). | |
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| Some other important properties of limits of real sequences include the following.
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| *The limit of a sequence is unique.
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| *<math>\lim_{n\to\infty} (a_n \pm b_n) = \lim_{n\to\infty} a_n \pm \lim_{n\to\infty} b_n</math>
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| *<math>\lim_{n\to\infty} c a_n = c \lim_{n\to\infty} a_n</math>
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| *<math>\lim_{n\to\infty} (a_n b_n) = (\lim_{n\to\infty} a_n)( \lim_{n\to\infty} b_n)</math>
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| *<math>\lim_{n\to\infty} \frac{a_n} {b_n} = \frac{ \lim_{n\to\infty} a_n}{ \lim_{n\to\infty} b_n}</math> provided <math>\lim_{n\to\infty} b_n \ne 0</math>
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| *<math>\lim_{n\to\infty} a_n^p = \left[ \lim_{n\to\infty} a_n \right]^p</math>
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| *If <math>a_n \leq b_n</math> for all <math>n</math> greater than some <math>N</math>, then <math>\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n </math>
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| *([[Squeeze Theorem]]) If <math>a_n \leq c_n \leq b_n</math> for all <math>n > N</math>, and <math>\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n = L</math>, {{pad|.5em}} then <math>\lim_{n\to\infty} c_n = L</math>.
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| *If a sequence is [[#Bounded|bounded]] and [[#Increasing and decreasing|monotonic]] then it is convergent.
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| *A sequence is convergent if and only if every subsequence is convergent.
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| These properties are extensively used to prove limits without the need to directly use the cumbersome formal definition. Once proven that <math>1/n \to 0</math> it becomes easy to show that <math>\frac{a}{b+c/n} \to \frac{a}{b}</math>, (<math>b \ne 0</math>), using the properties above.
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| ===Infinite limits===
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| The terminology and notation of convergence is also used to describe sequences whose terms become very large. A sequence <math>(x_n)</math> is said to '''tend to infinity''', written <math>x_n \to \infty</math> or <math>\lim_{n\to\infty}x_n = \infty</math> if, for every ''K'', there is an ''N'' such that, for every <math>n \geq N</math>, <math>x_n > K</math>; that is, the sequence terms are eventually larger than any fixed ''K''. Similarly, <math>x_n \to -\infty</math> if, for every ''K'', there is an ''N'' such that, for every <math>n \geq N</math>, <math>x_n < K</math>.
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| ==Metric spaces==
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| ===Definition===
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| A point ''x'' of the [[metric space]] (''X'', ''d'') is the '''limit''' of the [[sequence]] (''x<sub>n</sub>'') if, for all ε > 0, there is an ''N'' such that, for every <math>n \geq N</math>, <math>d(x_n, x) < \epsilon</math>. This coincides with the definition given for real numbers when <math>X = \mathbb{R}</math> and <math>d(x, y) = |x-y|</math>.
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| ===Properties===
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| For any [[continuous function]] ''f'', if <math>x_n \to x</math> then <math>f(x_n) \to f(x)</math>. In fact, a [[function (mathematics)|function]] ''f'' is continuous if and only if it preserves the limits of sequences.
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| Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for <math>\epsilon</math> less that half this distance, sequence terms cannot be within a distance <math>\epsilon</math> of both points.
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| ==Topological spaces==
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| ===Definition===
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| A point ''x'' of the topological space (''X'', τ) is the '''limit''' of the [[sequence]] (''x<sub>n</sub>'') if, for every [[topological neighbourhood|neighbourhood]] ''U'' of ''x'', there is an ''N'' such that, for every <math>n \geq N</math>, <math>x_n \in U</math>. This coincides with the definition given for metric spaces if (''X'',''d'') is a metric space and <math>\tau</math> is the topology generated by ''d''.
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| The limit of a sequence of points <math>\left(x_n:n\in \mathbb{N}\right)\;</math> in a topological space ''T'' is a special case of the [[Limit_of_a_function#Functions_on_topological_spaces|limit of a function]]: the domain is <math>\mathbb{N}</math> in the space <math>\mathbb{N} \cup \lbrace +\infty \rbrace</math> with the [[induced topology]] of the [[affinely extended real number system]], the range is ''T'', and the function argument ''n'' tends to +∞, which in this space is a [[limit point]] of <math>\mathbb{N}</math>.
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| ===Properties===
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| If ''X'' is a [[Hausdorff space]] then limits of sequences are unique where they exist. Note that this need not be the case in general; in particular, if two points ''x'' and ''y'' are [[topologically indistinguishable]], any sequence that converges to ''x'' must converge to ''y'' and vice-versa.
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| ==Cauchy sequences==
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| {{main|Cauchy sequence}}
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| [[File:Cauchy sequence illustration.svg|350px|thumb| The plot of a Cauchy sequence (''x<sub>n</sub>''), shown in blue, as ''x<sub>n</sub>'' versus ''n''. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as ''n'' increases. In the [[real numbers]] every Cauchy sequence converges to some limit.]]
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| A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in [[metric spaces]], and, in particular, in [[real analysis]]. One particularly important result in real analysis is ''Cauchy characterization of convergence for sequences'':
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| :A sequence is convergent if and only if it is Cauchy.
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| ==Definition in hyperreal numbers==
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| The definition of the limit using the [[hyperreal numbers]] formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence <math>(x_n)</math> tends to ''L'' if for every infinite [[hypernatural]] ''H'', the term ''x''<sub>''H''</sub> is infinitely close to ''L'', i.e., the difference ''x''<sub>''H''</sub> - ''L'' is [[infinitesimal]]. Equivalently, ''L'' is the [[Standard part function|standard part]] of ''x''<sub>''H''</sub>
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| :<math> L = {\rm st}(x_H)\,</math>.
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| Thus, the limit can be defined by the formula
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| :<math>\lim_{n \to H} x_n= {\rm st}(x_H),</math> | |
| where the limit exists if and only if the righthand side is independent of the choice of an infinite ''H''.
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| ==History==
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| The Greek philosopher [[Zeno of Elea]] is famous for formulating [[Zeno's paradoxes|paradoxes that involve limiting processes]].
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| [[Leucippus]], [[Democritus]], [[Antiphon (person)|Antiphon]], [[Eudoxus of Cnidus|Eudoxus]] and [[Archimedes]] developed the [[method of exhaustion]], which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a [[geometric series]].
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| [[Isaac Newton|Newton]] dealt with series in his works on ''Analysis with infinite series'' (written in 1669, circulated in manuscript, published in 1711), ''Method of fluxions and infinite series'' (written in 1671, published in English translation in 1736, Latin original published much later) and ''Tractatus de Quadratura Curvarum'' (written in 1693, published in 1704 as an Appendix to his ''Optiks''). In the latter work, Newton considers the binomial expansion of (''x''+''o'')<sup>''n''</sup> which he then linearizes by ''taking limits'' (letting ''o''→0).
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| In the 18th century, [[mathematician]]s like [[Leonhard Euler|Euler]] succeeded in summing some ''divergent'' series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, [[Joseph Louis Lagrange|Lagrange]] in his ''Théorie des fonctions analytiques'' (1797) opined that the lack of rigour precluded further development in calculus. [[Carl Friedrich Gauss|Gauss]] in his etude of [[hypergeometric series]] (1813) for the first time rigorously investigated under which conditions a series converged to a limit.
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| The modern definition of a limit (for any ε there exists an index ''N'' so that ...) was given by [[Bernhard Bolzano]] (''Der binomische Lehrsatz'', Prague 1816, little noticed at the time) and by Weierstrass in the 1870s.
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| ==See also==
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| *[[Limit of a function]]
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| *[[Net_%28mathematics%29#Limits_of_nets|Limit of a net]] — A [[net (mathematics)|net]] is a topological generalization of a sequence.
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| *[[Modes of convergence]]
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| * [[Richard Courant|Courant, Richard]] (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgoa.
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| * [[Frank Morley]] and [[James Harkness]] [http://www.archive.org/details/treatiseontheory00harkuoft A treatise on the theory of functions] (New York: Macmillan, 1893)
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| ==External links==
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| * {{springer|title=Limit|id=p/l058820}}
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| * [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html ''A history of the calculus'', including limits]
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| [[Category:Limits (mathematics)]]
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| [[Category:Sequences and series]]
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