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Sachin Kaushal, Lovely Professional University Unit 10: Limit of a Function

Unit 10: Limit of a Function Notes

CONTENTS
Objectives
Introduction

10.1 Notion of Limit
10.2 Sequential Limits
10.3 Algebra of Limits

10.4 Summary
10.5 Keywords
10.6 Review Questions
10.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Define limit of a function at a point and find its value
 Know sequential approach to limit of a function
 Find the limit of sum, difference, product and quotient of functions

Introduction

In earlier unit, we dealt with sequences and their limits. As you know, sequences are functions
whose domain is the set of natural numbers. In this unit, we discuss the limiting process for the
real functions with domains as subsets of the set R of real numbers and range also a subset of R.
What is the precise meaning for the intuitive idea of the values f(x) of a function f tending to or
approaching a number A as x approaches the number a? The search for an answer to this question
shall enable you to understand the concept of the limit which you have used in calculus. The
effect of algebraic operations of addition, subtraction, multiplication and division on the limits
of functions.

10.1 Notion of Limit

The intuitive idea of limit was used both by Newton and Leibnitz in their independent invention
of Differential Calculus around 1675. Later this notion of limit was also developed by D’Alembert.
“When the successive values attributed to a variable approach indefinitely a fixed value so as to
end by differing from it by as little as one wishes, this last is called the limit of all the others.”
Consider a simple example in which a function f is defined as
f(x) = 2x + 3, " x  R, x  1.

Give x the values which are near to 1 in the following way:
When x = 1.5, 1.4, 1.3, 1.2, 1.1, 1.01, 1.001

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