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The Rigorous Limit

Epsilon & Delta

A limit, made airtight. You name a tolerance ε around the target L; the limit exists only if there’s always a window δ around c keeping every nearby point inside that band.

shrink δ until the curve stays inside the gold band across the whole purple window
ƒ(x) ε-band (L ± ε) δ-window (c ± δ)