Limit

In mathematics the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a variable quantity approaches as closely as one desires. The operations of differentiation and integration from calculus are both based on the theory of limits. The theory of limits is based on a particular property of the real numbers; namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more.

Nearness is key to understanding limits: only after nearness is defined does a limit acquire an exact meaning. Relevantly, a neighborhood of points near any given point comprise a neighborhood. Neighborhoods are definitive components of infinite limits of a sequence.