Full Idea
When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others.
Gist of Idea
When successive variable values approach a fixed value, that is its 'limit'
Source
Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
Book Reference
Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.247
A Reaction
This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction?