Limit of a sequence

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, 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. 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 π r . {\displaystyle 2\pi r.} The corresponding sequence for inscribed polygons has the same limit.
nn sin(1/n)
10.841471
20.958851
...
100.998334
...
1000.999983

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

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