ABSTRACT
The concept of limit dates back to the late seventeenth century and the work of Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716). The subject of limits lacked mathematical rigor until 1821 when Augustin-Louis Cauchy (1789–1857) published his Cours d'Analyse in which he offered the following definition of limit: "If the successive values attributed to the same variable approach indefinitely a fixed value, such that finally they differ from it by as little as desired, this latter is called the limit of all the others." Based on the previous study of calculus, the student should have an intuitive notion of what it means for a function to be continuous. The chapter defines the limit at a point of a real-valued function defined on a subset of a metric space, and provide numerous examples to illustrate this idea. It then considers the closely related theory of continuity and investigate some of the consequences of this very important concept.
