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In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. [1] Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calculus problems; and notes for modern readers. [1]
Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol [51] Principles of Mathematical Analysis, by Walter Rudin [52] Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein [53] Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors [54]
The concept of limit was informally introduced for functions by Newton and Leibniz, at the end of the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy , and made rigorous, at the end of the 19th century by Bolzano and Weierstrass , who gave the modern ε-δ definition , which follows.