Limits, Continuity, and the Epsilon Idea

For every ε > 0 there exists δ > 0 such that for all x, if 0 < |x − a| < δ then |f(x) − L| < ε.

The skeptic challenges first with ε, then you respond; so δ is allowed to depend on ε (and on a), not the other way around.

It excludes x = a itself: the limit never depends on f(a), or even on whether f(a) exists — which is what lets us handle 0/0 difference quotients.

Assume δ ≤ 1 first to bound the nuisance factor (e.g. |x+3| < 7 when |x−3| < 1), then take δ = min{1, ε/bound}.

δ = min{1, ε/7}: then |x+3| < 7 and |x²−9| = |x−3||x+3| < (ε/7)(7) = ε.

There exists ε₀ > 0 such that for every δ > 0 there is an x with 0 < |x − a| < δ but |f(x) − L| ≥ ε₀.

Take ε₀ = 1/2; every interval (0, δ) contains points where sin(1/x) = 1 and points where it = −1, values 2 apart, so no L can be within 1/2 of both.

|x sin(1/x) − 0| ≤ |x|, so δ = ε works immediately — a disguised Squeeze Theorem argument.

If h(x) ≤ f(x) ≤ k(x) near a and both h and k tend to L as x → a, then f(x) → L as well.

Let ε > 0 be given; produce an explicit δ; assume 0 < |x − a| < δ; derive |f(x) − L| < ε.

Cauchy and Weierstrass in the 1800s; Berkeley mocked infinitesimals as 'the ghosts of departed quantities'.