What exactly is a Limit ?

\displaystyle\lim_{x\to a}f(x) = L \iff
\forall  \varepsilon >0, \exists \delta >0 such that
\boxed{0<|x-a|<\delta} \implies |f(x)-L|< \varepsilon

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 \delta = f (\varepsilon )
too counter-intuitive to grasp the idea by most university math students.

Intuitive Alternative:
\boxed { \displaystyle { \lim_{n \to \infty} P(n) = A} } \iff
Find any 2 natural numbers m (“Start“), k (“Scale“) such that:
\boxed { \text {For } m \leq n,  \frac {-k}{n} \leq P(n) - A \leq \frac {k}{n} }

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).


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