epsilon-delta definition

Saying f(x) 'approaches' L is intuitive, but a skeptic could ask: how close, and how close is close enough? The epsilon-delta definition answers with a challenge game between two players. A challenger names a tiny target tolerance around L — call it epsilon — and dares you to land f(x) within that tolerance. You answer by finding a tiny zone around a — call it delta — such that every input in your zone lands inside the challenger's target. If you can always meet any challenge, no matter how small, then the limit truly is L.

In symbols: lim x->a f(x) = L means that for every epsilon > 0 there exists a delta > 0 such that whenever 0 < |x - a| < delta, it follows that |f(x) - L| < epsilon. Unpacking the pieces: |f(x) - L| < epsilon says 'the output is within epsilon of L'; 0 < |x - a| < delta says 'the input is within delta of a, but not equal to a' — that little 0 < part is what lets us ignore the point a itself. The order matters: epsilon comes first (the demand), then you produce delta (your response, which is allowed to depend on epsilon).

This definition is what put calculus on solid ground in the 1800s, replacing vague talk of 'infinitely small' quantities with something you can actually prove. You will not compute everyday limits this way — algebra and the limit laws handle that. Its real job is to make 'approaches' precise enough that theorems can be proved without hand-waving. Continuity is just this same definition with L replaced by f(a).