CHAPTER IV - Limits and Continuity

Abstract

This chapter discusses the concept of limits and continuity. The concept of limit plays a fundamental part in the calculus. The chapter describes sequence as an infinite sequence whose terms are real numbers. The limit of a sequence may not be equal to any of its terms. The chapter also describes divergent sequences. These are of two types: (1) those diverging to ±∞, and (2) those that oscillate. It also provides an overview of theorems on limits. It would be tedious if each problem on limits had to be solved by appeal to the appropriate definition. General theorems enable solving problems more efficiently. The chapter discusses plane curve. The notion of continuity enables in understanding the concept of a plane curve. This concept is explained in terms of the motion of a particle moving in the plane. Such a motion is specified by taking a rectangular coordinate system in the plane, and stating the coordinates (x, y) of the positions occupied by the particle at different instances of time t—that is, by giving two functions f and g, whose arguments are certain real numbers t and whose values f(t) and g(t) give the position coordinates of the particle at instant t.