Limit and Continuity

Abstract

In this chapter, we introduce the concept of the limit of a function. This concept bridges the gap between the areas of algebra, geometry, and calculus. The limit is the most important concept of calculus, without limits calculus simply does not exist. Practically, every notion of calculus is a limit in one sense or another. Physical concepts and real-world situations can be expressed as limits of certain functions. For example, a circle is the limit of a polygon, the length of a curve is the limit of the lengths of polygonal paths, the area of a region bounded by a curve is the limit of the sum of areas of approximating rectangles, and instantaneous velocity (velocity at a particular time) is the limit of average velocities. We also discuss the concept of continuous and discontinuous functions in terms of limit.