Editing Limit of a sequence (section)

{{Short description|Value to which tends an infinite sequence}}
{{For|the general mathematical concept|Limit (mathematics)}}
{{more citations needed|date=May 2017}}

[[File:Archimedes pi.svg|350px|right|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle|The sequence given by the perimeters of regular ''n''-sided [[polygon]]s that [[circumscribe]] the [[unit circle]] has a limit equal to the perimeter of the circle, i.e. <math>2\pi</math>. The corresponding sequence for inscribed polygons has the same limit.]]
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!<math>n</math>!!<math>n\times \sin\left(\tfrac1{n}\right)</math>
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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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As the positive [[integer]] <math display="inline">n</math> becomes larger and larger, the value <math display="inline">n\times \sin\left(\tfrac1{n}\right)</math> becomes arbitrarily close to <math display="inline">1</math>. We say that "the limit of the sequence <math display="inline">n \times \sin\left(\tfrac1{n}\right)</math> equals <math display="inline">1</math>."
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In [[mathematics]], the '''limit of a sequence''' is the value that the terms of a [[sequence]] "tend to", and is often denoted using the <math>\lim</math> symbol (e.g., <math>\lim_{n \to \infty}a_n</math>).<ref name="Courant (1961), p. 29">Courant (1961), p. 29.</ref> If such a limit exists and is finite, the sequence is called '''convergent'''.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Convergent Sequence|url=https://mathworld.wolfram.com/ConvergentSequence.html|access-date=2020-08-18|website=mathworld.wolfram.com|language=en}}</ref> A sequence that 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]] ultimately rests.<ref name="Courant (1961), p. 29"/>

Limits can be defined in any [[metric space|metric]] or [[topological space]], but are usually first encountered in the [[real number]]s.