limit (concept)

A limit is the value a process homes in on, like the number a thermostat settles toward without ever perfectly touching, or the destination a traveller approaches by halving the remaining distance again and again. The limit is the answer the approach points to — whether or not the process ever arrives there.

More precisely, a limit is a value L approached as closely as desired by some varying quantity, where ‘as closely as desired’ is cashed out by the epsilon–delta (or epsilon–N) criterion. The quantity may be the terms of a sequence, the values of a function near a point, partial sums of a series, or many other things; in each case the limit is the single value that the epsilon-game can always reach.

Two honest cautions. First, the value of a limit need not be attained: the limit of 1/n is 0 even though 1/n is never 0. Second, a limit, if it exists, is unique — but it may fail to exist (the sequence (-1)^n approaches no single value). The limit is the load-bearing idea of all analysis; differentiation, integration, continuity, infinite series and the real numbers themselves are defined through it.