if, for every real number , there exists a real number such that, whenever with , then .
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 spaces, under some mild restrictions. In this context we say that if is a limit point of and, for every neighborhood of (in ), there is a deleted neighborhood of (in ) which is mapped into by . One also requires that the range be Hausdorff (or at least ) in order to ensure that limits, when they exist, are unique.
The definition of the limit of a mapping can be based on the limit of a sequence. To wit, if and only if, for every sequence of points in converging to (that is, , ), the sequence of points in converges to .
|Date of creation||2013-03-22 12:28:25|
|Last modified on||2013-03-22 12:28:25|
|Last modified by||djao (24)|