Limit of a sequence

For the general mathematical concept, see Limit (mathematics).

The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. $2\pi$. The corresponding sequence for inscribed polygons has the same limit.

$n$	$n\times \sin \left({\tfrac {1}{n}}\right)$
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., $\lim _{n\to \infty }a_{n}$). If such a limit exists and is finite, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

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