Let X and Y be metric spaces and let aX be a limit pointPlanetmathPlanetmath of X. Suppose that f:X{a}Y is a function defined everywhere except at a. For LY, we say the limit of f(x) as x approaches a is equal to L, or


if, for every real number ε>0, there exists a real number δ>0 such that, whenever xX with 0<dX(x,a)<δ, then dY(f(x),L)<ε.

The formal definition of limit as given above has a well–deserved reputation for being notoriously hard for inexperienced students to master. There is no easy fix for this problem, since the concept of a limit is inherently difficult to state precisely (and indeed wasn’t even accomplished historically until the 1800’s by Cauchy, well after the development of calculus in the 1600’s by Newton and Leibniz). However, there are number of related definitions, which, taken together, may shed some light on the nature of the concept.

  • The notion of a limit can be generalized to mappings between arbitrary topological spacesMathworldPlanetmath, under some mild restrictionsPlanetmathPlanetmathPlanetmathPlanetmath. In this context we say that limxaf(x)=L if a is a limit point of X and, for every neighborhoodMathworldPlanetmathPlanetmath V of L (in Y), there is a deleted neighborhood U of a (in X) which is mapped into V by f. One also requires that the range Y be HausdorffPlanetmathPlanetmath (or at least T1) in order to ensure that limits, when they exist, are unique.

  • Let an,n be a sequence of elements in a metric space X. We say that LX is the limit of the sequence, if for every ε>0 there exists a natural numberMathworldPlanetmath N such that d(an,L)<ε for all natural numbers n>N.

  • The definition of the limit of a mapping can be based on the limit of a sequence. To wit, limxaf(x)=L if and only if, for every sequence of points xn in X converging to a (that is, xna, xna), the sequence of points f(xn) in Y converges to L.

In calculus, X and Y are frequently taken to be Euclidean spaces n and m, in which case the distance functions dX and dY cited above are just Euclidean distance.

Title limit
Canonical name Limit
Date of creation 2013-03-22 12:28:25
Last modified on 2013-03-22 12:28:25
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 26A06
Classification msc 26B12
Classification msc 54E35
Related topic ContinuousPlanetmathPlanetmath