Limit

The idea that a sequence or function can approach a fixed value as closely as desired without necessarily reaching it, the foundation on which the derivative and the integral of calculus both rest. Its root lies in the ancient Greek method of exhaustion, put on a systematic basis by Eudoxus around 370 BC and used with great skill by Archimedes to find areas and volumes by a chain of reasoning that avoided any appeal to infinity or infinitesimals directly. Isaac Newton and Gottfried Leibniz built the calculus of the seventeenth century on an intuitive, unrigorous notion of quantities becoming infinitely small, which stood for over a century before Augustin-Louis Cauchy gave a rigorous verbal definition in his 1821 Cours d'Analyse and Karl Weierstrass later completed the modern epsilon-delta treatment, removing any need for infinitesimals at all.