AP Calculus AB and BC glossary

Epsilon-delta definition

Also called: Formal definition of a limit

The epsilon-delta definition states that a limit equals L if, for every tolerance around L, there is a distance around the input point that keeps all outputs within that tolerance. It is the precise version of the informal idea of approaching a value.

∀ε>0, ∃δ>0: 0<∣x−a∣<δ  ⟹  ∣f(x)−L∣<ε\forall \varepsilon > 0, \ \exists \delta > 0: \ 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon

Read it as a challenge and a response: someone names a tolerance, and you must produce a closeness that guarantees it. The limit exists when you can always answer, no matter how small the tolerance.

The condition 0<∣x−a∣0 < |x - a| deliberately excludes the point itself, which is the formal statement of the idea that a limit ignores the function's value at the point.

On the AP exam

You are expected to understand what the definition says, not to construct epsilon-delta proofs. The computational work is done with limit laws and algebra.

Appears in: Unit 1: Limits and Continuity