Limit

The limit of a function $f$ at a point a is L, written $\lim_{x\to a} f(x) = L$, when the values of f can be kept as close to L as anyone asks by keeping x close enough to a — without ever looking at x = a itself. The precise form is the ε–δ definition (Cauchy and Weierstrass): a challenge ε and a response δ.

On the graph the challenge is the red band of half-width ε round the blue line y = L; its dashed edges are not part of it, since the inequality is strict. The response is the yellow window of half-width δ round a, with a gap at a: the condition $0 < |x - a|$ leaves x = a out, which is why a hole there, or a value f(a) defined elsewhere, changes nothing. The window works when the curve inside it stays inside the band; when it does not, a yellow × marks a point that escapes. A one-sided limit (x → a⁻ or a⁺) uses half the window. The limit, when it exists, is unique: two values cannot both fit inside every band.

The page reports two different δ. The guaranteed δ comes from an inequality that holds at every x in the window, so it is a proof: for x², the hole, the jump, sin(1/x) and 1/x² the largest δ has a closed form and is itself the guarantee; for x·sin(1/x) the bound $|x\sin(1/x)| \le |x|$ gives δ = ε at 0, and for sin x / x the bound $1 - x^2/6 \le \sin x / x < 1$ gives $\delta = \sqrt{6\varepsilon}$. For those two functions the largest δ is numerical: the curve is followed piece by piece between its turning points, and each crossing of the band's edge is found by bisection. Any smaller δ works as well; δ is never unique.