The above scary ‘epsilon-delta’ definition of “Limit” by the French mathematician Cauchy in 19th century is the standard rigorous definition in today’s Analysis textbooks.
It was not taught in my Cambridge GCE A-Level Pure Math in 1970s (still true today), but every French Baccalaureate Math student (Terminale, equivalent to JC 2 or Pre-U 2) knows it by heart. A Cornell University Math Dean recalled how he was told by his high-school teacher to memorise it — even though he did not fully understand — the “epsilon-delta” definition by “chanting”:
“for all epsilon, there is a delta ….”
(French: Quelque soit epsilon, il existe un delta …)
In this video, I am glad someone like Prof N. Wildberger recognised its “flaws” albeit rigorous, by suggesting another more intuitive definition:
◇ Cauchy’s “flaw”: ambiguous
Finding a certain
too counter-intuitive to grasp the idea by most university math students.
◇ Intuitive Alternative:
Find any 2 natural numbers m (“Start“), k (“Scale“) such that:
A Simple Analogy in Life:
Let P(n) = Any Person’s lifespan of age n
m = ‘Start’ Age to retire, say 60
k = ‘Scale’ of interval (in years, eg. 1 year)
A = Limit of a person’s lifespan, say 80 (male) or 85 (female)
As we grow older (n increases), from a certain “Start” point (m), our lifespan P(n) approaches the limit A, plus or minus k/n (years).