BMTC - 131
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
2
LIMIT AND CONTINUITY
Block Introduction 3
Notations and Symbols 4
UNIT 6
Real Numbers 5
UNIT 7
Limit 47
UNIT 8
Continuity 95
Miscellaneous Examples and Exercises 121
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and the UGC-
CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to Sh.
Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of this
block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-39-3
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
2 Prof. M. S. Nathawat, Director, School of Sciences.
,BLOCK 2 LIMIT AND CONTINUITY
This is the second of the five blocks which you will be studying for the course calculus. We
assume that you are familiar with the real number system and real functions. But, just to
refresh your memory, we have given a brief account of real numbers and their properties,
as well as some types of functions in Unit 6. It is also possible that some of you have not
studied certain aspects of the real number system and functions earlier. In that case, Unit
6 will help you prepare a firm ground for the imposing structure of calculus which follows.
Calculus has two fundamental procedures, differentiation and integration, which can be
formulated in terms of a concept called the ‘limit’. In Unit 7, we begin with helping you
acquire an intuitive sense of this concept. The word ‘intuitive’ can mean several things. Its
use here means “experience based, without proof”. Following this intuitive presentation,
we present the formal definition of ‘limit’. In addition, we will introduce you to functions that
require the use of limits, namely, the exponential, logarithmic functions and hyperbolic
functions.
In Unit 8, we will continue to use ‘limit’ to explore a new concept, namely, that of
continuity. We will also discuss the types of discontinuity, and end the unit by stating the
intermediate value theorem for continuous functions, and giving some of its applications.
In Unit 6 to Unit 8, we have included a number of examples. Please go through them
carefully. They will help you in a better understanding of the concepts discussed and will
also serve as a guide in solving the exercises.
At the end of the block, you will find miscellaneous examples and exercises, covering the
concepts you have studied across the units. Please solve the exercises on your own. At
the end of each unit, and after the miscellaneous exercises, we do provide some
solutions/answers to the exercises concerned. These are only as a support for you to be
able to check whether you have been able to solve the problem correctly or not. Please do
not look at these solutions till you have spend enough time on studying the unit and trying
all the exercises.
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 2)
∈ (∉) belongs to (does not belong to)
N the set of natural numbers
Z (Z + )(Z − ) the set of integers(the set of positive integers)(the set of negative
integers)
( )
Q Q* the set of rational numbers (non-zero rational numbers)
R (R )(R )
+ −
the set of real numbers(the set of positive real numbers)(the set of
negative real numbers)
⇒ (⇔ ) implies (implies and is implied by)
iff if and only if
∴ therefore
w.r.t. with respect to
s.t. such that
< (≤) is less than (is less than or equal to)
> (≥) is greater than (is greater than or equal to)
∃ there exists
∀ for all
f :X → Y f is a function from the set X to the set Y
{x | x satisfies P} the set of all x such that x satisfies the property P
|x| modulus of the real x
lim f (x ) limit of f ( x ) as x tends to a
x→a
x → f (x) a functions f taking x to f ( x )
n
Cr the number of combinations of r things taken out of n ,
n n!
Cr =
r !(n − r) !
≈ is approximately equal to
max {x, y} the maximum of x and y
min {x , y} the minimum of x and y
Please see the notations and symbols used in Block 1.
4
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
2
LIMIT AND CONTINUITY
Block Introduction 3
Notations and Symbols 4
UNIT 6
Real Numbers 5
UNIT 7
Limit 47
UNIT 8
Continuity 95
Miscellaneous Examples and Exercises 121
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and the UGC-
CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to Sh.
Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of this
block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-39-3
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
2 Prof. M. S. Nathawat, Director, School of Sciences.
,BLOCK 2 LIMIT AND CONTINUITY
This is the second of the five blocks which you will be studying for the course calculus. We
assume that you are familiar with the real number system and real functions. But, just to
refresh your memory, we have given a brief account of real numbers and their properties,
as well as some types of functions in Unit 6. It is also possible that some of you have not
studied certain aspects of the real number system and functions earlier. In that case, Unit
6 will help you prepare a firm ground for the imposing structure of calculus which follows.
Calculus has two fundamental procedures, differentiation and integration, which can be
formulated in terms of a concept called the ‘limit’. In Unit 7, we begin with helping you
acquire an intuitive sense of this concept. The word ‘intuitive’ can mean several things. Its
use here means “experience based, without proof”. Following this intuitive presentation,
we present the formal definition of ‘limit’. In addition, we will introduce you to functions that
require the use of limits, namely, the exponential, logarithmic functions and hyperbolic
functions.
In Unit 8, we will continue to use ‘limit’ to explore a new concept, namely, that of
continuity. We will also discuss the types of discontinuity, and end the unit by stating the
intermediate value theorem for continuous functions, and giving some of its applications.
In Unit 6 to Unit 8, we have included a number of examples. Please go through them
carefully. They will help you in a better understanding of the concepts discussed and will
also serve as a guide in solving the exercises.
At the end of the block, you will find miscellaneous examples and exercises, covering the
concepts you have studied across the units. Please solve the exercises on your own. At
the end of each unit, and after the miscellaneous exercises, we do provide some
solutions/answers to the exercises concerned. These are only as a support for you to be
able to check whether you have been able to solve the problem correctly or not. Please do
not look at these solutions till you have spend enough time on studying the unit and trying
all the exercises.
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 2)
∈ (∉) belongs to (does not belong to)
N the set of natural numbers
Z (Z + )(Z − ) the set of integers(the set of positive integers)(the set of negative
integers)
( )
Q Q* the set of rational numbers (non-zero rational numbers)
R (R )(R )
+ −
the set of real numbers(the set of positive real numbers)(the set of
negative real numbers)
⇒ (⇔ ) implies (implies and is implied by)
iff if and only if
∴ therefore
w.r.t. with respect to
s.t. such that
< (≤) is less than (is less than or equal to)
> (≥) is greater than (is greater than or equal to)
∃ there exists
∀ for all
f :X → Y f is a function from the set X to the set Y
{x | x satisfies P} the set of all x such that x satisfies the property P
|x| modulus of the real x
lim f (x ) limit of f ( x ) as x tends to a
x→a
x → f (x) a functions f taking x to f ( x )
n
Cr the number of combinations of r things taken out of n ,
n n!
Cr =
r !(n − r) !
≈ is approximately equal to
max {x, y} the maximum of x and y
min {x , y} the minimum of x and y
Please see the notations and symbols used in Block 1.
4
