TABLE OF CONTENTS Soon after geometry became a separate branch of mathematics, geometers began to study problems they could best solve by processes that they imagined could be continued indefinitely. These were problems related to vanishingly small distances, and problems related to sets containing infinitely many elements. For example, a circle begins to coincide with the limiting figure of a regular polygon inscribed in it as we increase the number of sides, giving rise to the concept of a nonending process. Or, we can successively halve the distance between two points AB on a straight line segment to produce lengths 1/2, 1/4, 1/8, , 1/2 n that of the initial length, again setting off a nonending process and introducing vanishingly small distances. Furthermore, the set of points on a line segment is an infinite set. All of these processes are related to the continuity of line segments and to the concept of limit. These, in turn, provide the basis for the calculus.
Limit and continuity processes are at work in computer graphics applications to ensure the display of smooth, continuous-looking curves and surfaces. They are an important consideration in the construction and rendering of shapes for computer-aided design and manufacturing, scientific visualization, and similar applications.
This chapter explains limit and continuity. It begins with the Greek method of exhaustion, which appeals strongly to our intuition and sets the stage for a more rigorous discussion of the limit process to come. The section on sequences and series extends this appeal...
