Limit of a sequence

$n$	$n\times \sin \left({\tfrac {1}{n}}\right)$
1	0.841471
2	0.958851
...
10	0.998334
...
100	0.999983

As the positive integer ${\textstyle n}$ becomes larger and larger, the value ${\textstyle n\times \sin \left({\tfrac {1}{n}}\right)}$ becomes arbitrarily close to ${\textstyle 1}$ . We say that "the limit of the sequence ${\textstyle n\times \sin \left({\tfrac {1}{n}}\right)}$ equals ${\textstyle 1}$ ."

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the $\lim$ symbol (e.g., $\lim _{n\to \infty }a_{n}$ ).^[1] If such a limit exists, the sequence is called convergent.^[2] A sequence that does not converge is said to be divergent.^[3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.^[1]

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

^ ^a ^b Courant (1961), p. 29.
^ Weisstein, Eric W. "Convergent Sequence". mathworld.wolfram.com. Retrieved 2020-08-18.
^ Courant (1961), p. 39.

[Courant_(1961),_p._29-1] Courant (1961), p. 29.

[2] Weisstein, Eric W. "Convergent Sequence". mathworld.wolfram.com. Retrieved 2020-08-18.

[3] Courant (1961), p. 39.

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