This document provides an overview of limits and continuity in mathematics. It begins by introducing functions of discrete and continuous variables, and defines the concept of a limit for sequences. A limit describes the long-term behavior of a function as the input approaches some value. The document then discusses finding the limit of various sequences, such as sequences that approach 0. It also introduces subsequences and defines the algebra of limits of sequences, such as how the limit behaves under operations like addition, subtraction, multiplication and division. The overall goal is to understand the concept of limit and apply it to determine if functions are continuous or not.
The document provides an overview of topics in number theory including:
- Number systems such as natural numbers, integers, and real numbers
- Properties of real numbers like closure, commutativity, associativity, identity, and inverse properties
- Rational and irrational numbers
- Order of operations
- Absolute value
- Intervals on the number line
- Finite and repeating decimals
- Converting between fractions and decimals
This section provides an overview of elementary number theory concepts including:
- Divisibility and prime/composite numbers
- The division theorem and remainders
- Common divisors and the greatest common divisor (GCD)
- Euclid's algorithm for efficiently calculating the GCD of two numbers in O(logN) time
- Modular arithmetic and properties used to perform arithmetic operations modulo a number for large inputs.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
The document discusses the evolution and properties of number systems. It begins by describing early counting systems used by ancient civilizations like Egyptians. It then summarizes the step-by-step development of more advanced number systems like integers, rational numbers, and real numbers to solve increasingly complex equations. Key properties of real numbers such as closure, identity, inverse, commutativity, associativity and distributivity are defined. Rational and irrational numbers are distinguished based on their representation as terminating/recurring versus non-terminating and non-recurring decimals.
The document summarizes the development of number systems from ancient times to modern real numbers. It begins with early humans using tally marks to count sheep and cattle. Ancient Egyptians had a base-10 system using symbols for 10 and 100 that were repeated to represent larger numbers. Over time, more advanced societies developed more complex systems until the modern real number system was established. This system includes natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses the properties of these different types of numbers.
I am Martin J. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document provides information about several advanced algebra topics:
1. The Remainder Theorem states that when dividing a polynomial f(x) by x-c, the remainder is equal to f(c).
2. The Factor Theorem states that if f(c) = 0, then x-c is a factor of the polynomial.
3. Synthetic division is a method for dividing polynomials without subtracting, by multiplying the coefficients.
4. The Rational Zeros Theorem can be used to find all possible rational zeros of a polynomial by considering factors of the leading coefficient and constant term.
This document provides information on number theory topics covered in Unit 1 of a mathematics course, including:
- Number systems such as natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
- Representing rational and irrational numbers on the number line. Rational numbers are easy to represent precisely, while irrational numbers correspond to points.
- Properties of finite and repeating decimals, including how to write decimals as fractions.
- Absolute value, which is the distance from a number to zero on the number line.
The document provides an overview of topics in number theory including:
- Number systems such as natural numbers, integers, and real numbers
- Properties of real numbers like closure, commutativity, associativity, identity, and inverse properties
- Rational and irrational numbers
- Order of operations
- Absolute value
- Intervals on the number line
- Finite and repeating decimals
- Converting between fractions and decimals
This section provides an overview of elementary number theory concepts including:
- Divisibility and prime/composite numbers
- The division theorem and remainders
- Common divisors and the greatest common divisor (GCD)
- Euclid's algorithm for efficiently calculating the GCD of two numbers in O(logN) time
- Modular arithmetic and properties used to perform arithmetic operations modulo a number for large inputs.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
The document discusses the evolution and properties of number systems. It begins by describing early counting systems used by ancient civilizations like Egyptians. It then summarizes the step-by-step development of more advanced number systems like integers, rational numbers, and real numbers to solve increasingly complex equations. Key properties of real numbers such as closure, identity, inverse, commutativity, associativity and distributivity are defined. Rational and irrational numbers are distinguished based on their representation as terminating/recurring versus non-terminating and non-recurring decimals.
The document summarizes the development of number systems from ancient times to modern real numbers. It begins with early humans using tally marks to count sheep and cattle. Ancient Egyptians had a base-10 system using symbols for 10 and 100 that were repeated to represent larger numbers. Over time, more advanced societies developed more complex systems until the modern real number system was established. This system includes natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses the properties of these different types of numbers.
I am Martin J. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
This document provides information about several advanced algebra topics:
1. The Remainder Theorem states that when dividing a polynomial f(x) by x-c, the remainder is equal to f(c).
2. The Factor Theorem states that if f(c) = 0, then x-c is a factor of the polynomial.
3. Synthetic division is a method for dividing polynomials without subtracting, by multiplying the coefficients.
4. The Rational Zeros Theorem can be used to find all possible rational zeros of a polynomial by considering factors of the leading coefficient and constant term.
This document provides information on number theory topics covered in Unit 1 of a mathematics course, including:
- Number systems such as natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
- Representing rational and irrational numbers on the number line. Rational numbers are easy to represent precisely, while irrational numbers correspond to points.
- Properties of finite and repeating decimals, including how to write decimals as fractions.
- Absolute value, which is the distance from a number to zero on the number line.
We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
The document discusses recursive definitions of sequences, functions, sets, and strings. It provides examples of recursively defining the Fibonacci sequence, factorial function, set of prices using quarters and dimes, and set of binary numbers. It also discusses recursively defining the length, empty string, concatenation, and reversal of strings.
The document provides definitions and explanations of key concepts in algebra including:
1. Types of numbers such as complex, rational, irrational, and integer numbers.
2. Properties of real numbers like commutative, associative, and distributive properties.
3. Exponents, radicals, logarithms, progressions, the binomial theorem, and word problems.
This document provides an overview of key concepts in calculus, including limits, derivatives, and continuity. It discusses how limits describe the behavior of a function as its input approaches a certain value. Specific topics covered include tangent lines and limits, areas and limits, decimals and limits, one-sided limits, infinite limits, vertical asymptotes, computing limits using limit laws, and continuity of functions. The document also introduces derivatives as a new function whose value at each x is the derivative of the original function at that point.
I am Martin J. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
Recursive Definitions in Discrete Mathmatcs.pptxgbikorno
The document discusses recursive definitions, which define an object in terms of itself. It provides examples of recursively defined sequences, functions, and sets. Recursion is related to mathematical induction. Recursive algorithms solve problems by reducing them to smaller instances of the same problem. While recursive definitions and algorithms are elegant, iterative equivalents are typically more efficient in terms of time and space usage.
This document discusses recursive functions and growth functions. It defines recursive functions as functions defined in terms of previous values using initial conditions and recurrence relations. Examples of recursively defined sequences like Fibonacci are provided. Growth functions are defined using big-O notation to analyze how functions grow relative to each other. Common proof techniques like direct proof, indirect proof, proof by contradiction and induction are described. Walks and paths in trees are defined as sequences of alternating vertices and edges that begin and end at vertices. Deterministic finite automata are defined as 5-tuples with states, input alphabet, transition function, start state, and set of accepting/final states.
The document discusses various topics related to sequences including:
- Definitions of sequences and different types of sequences such as arithmetic progressions, geometric progressions, and recurrence relations.
- Examples of sequences used in computer programming to determine if a number is even or odd through modulo operations.
- How the principle of mathematical induction can be used to prove statements about sequences, including the first and second principles of mathematical induction.
The document discusses functions and algorithms. It defines what a function is and provides examples. It also discusses different types of functions such as one-to-one, onto, and inverse functions. The document then discusses algorithms and complexity analysis. It provides examples of linear search algorithms and analyzes their worst case and average case time complexities.
1) The document discusses calculus concepts of derivatives and integrals. Derivatives measure the rate of change of a function, while integrals calculate the area under a function.
2) It provides examples of how to calculate derivatives and integrals, including the rules for derivatives of constants, powers, sums, products, and compositions. Integrals are calculated by dividing the area into rectangles.
3) As an application, it shows how to calculate the economic order quantity by setting the derivative of the total cost function equal to zero to find the minimum.
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
The real number system contains rational numbers like integers and fractions, as well as irrational numbers like the square root of 2. It is a complete system where every non-empty set that is bounded above has a least upper bound. This property of completeness is what allows real numbers to correspond precisely to points on a number line. The Archimedean property, which follows from completeness, states that between any real number and zero there is a positive integer multiple. This implies that rational numbers are dense within the real numbers.
This document discusses sequences and their limits. Some key points:
- A sequence is a list of numbers written in a definite order. It can be thought of as a function with domain the positive integers.
- The limit of a sequence is defined similar to the limit of a function. The limit of a sequence {an} as n approaches infinity is L if the terms can be made arbitrarily close to L by making n sufficiently large.
- A sequence is convergent if it approaches a finite limit. It is divergent if the terms approach infinity. Bounded monotonic sequences are always convergent due to the completeness of real numbers.
1) The document discusses the real number system, including identifying integers, rational numbers, and irrational numbers. Real numbers have important properties like closure under addition and multiplication.
2) Intervals of real numbers can be described using inequality notation like x ≥ 1 or -4 < x < 1, or interval notation like [1, +∞) or (-4, 1).
3) Absolute value of a real number a, written |a|, represents the distance of the number from the origin on the number line and is always non-negative.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
The document discusses recursive definitions of sequences, functions, sets, and strings. It provides examples of recursively defining the Fibonacci sequence, factorial function, set of prices using quarters and dimes, and set of binary numbers. It also discusses recursively defining the length, empty string, concatenation, and reversal of strings.
The document provides definitions and explanations of key concepts in algebra including:
1. Types of numbers such as complex, rational, irrational, and integer numbers.
2. Properties of real numbers like commutative, associative, and distributive properties.
3. Exponents, radicals, logarithms, progressions, the binomial theorem, and word problems.
This document provides an overview of key concepts in calculus, including limits, derivatives, and continuity. It discusses how limits describe the behavior of a function as its input approaches a certain value. Specific topics covered include tangent lines and limits, areas and limits, decimals and limits, one-sided limits, infinite limits, vertical asymptotes, computing limits using limit laws, and continuity of functions. The document also introduces derivatives as a new function whose value at each x is the derivative of the original function at that point.
I am Martin J. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Minnesota, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
Recursive Definitions in Discrete Mathmatcs.pptxgbikorno
The document discusses recursive definitions, which define an object in terms of itself. It provides examples of recursively defined sequences, functions, and sets. Recursion is related to mathematical induction. Recursive algorithms solve problems by reducing them to smaller instances of the same problem. While recursive definitions and algorithms are elegant, iterative equivalents are typically more efficient in terms of time and space usage.
This document discusses recursive functions and growth functions. It defines recursive functions as functions defined in terms of previous values using initial conditions and recurrence relations. Examples of recursively defined sequences like Fibonacci are provided. Growth functions are defined using big-O notation to analyze how functions grow relative to each other. Common proof techniques like direct proof, indirect proof, proof by contradiction and induction are described. Walks and paths in trees are defined as sequences of alternating vertices and edges that begin and end at vertices. Deterministic finite automata are defined as 5-tuples with states, input alphabet, transition function, start state, and set of accepting/final states.
The document discusses various topics related to sequences including:
- Definitions of sequences and different types of sequences such as arithmetic progressions, geometric progressions, and recurrence relations.
- Examples of sequences used in computer programming to determine if a number is even or odd through modulo operations.
- How the principle of mathematical induction can be used to prove statements about sequences, including the first and second principles of mathematical induction.
The document discusses functions and algorithms. It defines what a function is and provides examples. It also discusses different types of functions such as one-to-one, onto, and inverse functions. The document then discusses algorithms and complexity analysis. It provides examples of linear search algorithms and analyzes their worst case and average case time complexities.
1) The document discusses calculus concepts of derivatives and integrals. Derivatives measure the rate of change of a function, while integrals calculate the area under a function.
2) It provides examples of how to calculate derivatives and integrals, including the rules for derivatives of constants, powers, sums, products, and compositions. Integrals are calculated by dividing the area into rectangles.
3) As an application, it shows how to calculate the economic order quantity by setting the derivative of the total cost function equal to zero to find the minimum.
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
The real number system contains rational numbers like integers and fractions, as well as irrational numbers like the square root of 2. It is a complete system where every non-empty set that is bounded above has a least upper bound. This property of completeness is what allows real numbers to correspond precisely to points on a number line. The Archimedean property, which follows from completeness, states that between any real number and zero there is a positive integer multiple. This implies that rational numbers are dense within the real numbers.
This document discusses sequences and their limits. Some key points:
- A sequence is a list of numbers written in a definite order. It can be thought of as a function with domain the positive integers.
- The limit of a sequence is defined similar to the limit of a function. The limit of a sequence {an} as n approaches infinity is L if the terms can be made arbitrarily close to L by making n sufficiently large.
- A sequence is convergent if it approaches a finite limit. It is divergent if the terms approach infinity. Bounded monotonic sequences are always convergent due to the completeness of real numbers.
1) The document discusses the real number system, including identifying integers, rational numbers, and irrational numbers. Real numbers have important properties like closure under addition and multiplication.
2) Intervals of real numbers can be described using inequality notation like x ≥ 1 or -4 < x < 1, or interval notation like [1, +∞) or (-4, 1).
3) Absolute value of a real number a, written |a|, represents the distance of the number from the origin on the number line and is always non-negative.
Similar to Unit 05 - Limits and Continuity.pdf (20)
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
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Answers about how you can do more with Walmart!"
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
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ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
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at Integral University, Lucknow, 06.06.2024
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1. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 98
Unit 5 Limits and Continuity
Structure:
5.1 Introduction
Objectives
5.2 The Real Number System
5.3 The Concept of Limit
5.4 Concept of Continuity
5.5 Summary
5.6 Terminal Questions
5.7 Answers
5.1 Introduction
In this chapter you will be recalling the properties of number. You will be
studying the limits of a function of a discrete variables, represented as a
sequence and the limit of functions of a real variables. Both these limits
describe the long term behaviour of functions. You will be studying
continuity which is essential for describing a process that goes on without
abrupt changes. You will see a good number of examples for understanding
the concepts clearly. As mathematics is mastered only by doing, examples
are given for practice.
You are familiar with numbers and using them in day – to – day life. Before
introducing the concept of limits let us refresh our memory regarding various
types of numbers.
Objectives:
At the end of the unit you would be able to
understand the concept of limit.
apply the concept of continuity in problems.
find whether a given function is continuous or not.
5.2 The Real Number System
You are using numbers like i
3
2
,
i
1
,
i
,
i
,
,
7
4
,
4
3
,
3
,
2
etc.
The last two numbers 1 + i and 2 – 3i are complex numbers. The rest of
them are real numbers.
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The numbers 1, 2, 3, …… are called natural numbers.
N = {1, 2, 3, ……..}
The numbers …………………….. – 3, –2, –1, 0, 1, 2, 3, …… are integers.
,........
2
,
1
,
0
Z
or
..........
..........
,
3
,
2
,
1
,
0
,
1
,
2
,
3
.....,
The set of quotients of two integers, the denominator not equal to 0 are
called rational numbers and the set of rational numbers is denoted by Q.
Usually these numbers are represented as points on a horizontal line called
the real axis. (Refer to Fig. 5.1)
Fig. 5.1 Representation of numbers
After representing the integers and rational numbers. So there are no gaps
in the real line and so it is called “continuous”.
We can also represent the relations “greater than” or “less than”
geometrically. If a < b, then a lies to the left of b in the real line (and b lies to
the right of a).
The modulus function
The modulus function simply represents the numerical value of a number. It
is defined as follows:
0
x
if
x
0
x
if
x
x
For example, 0
0
,
2
2
,
2
2
Note: b
a
b
a
7
4
– 3 – 2 – 1 0 1 2 3
4
3
2
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SAQ: (Self Assessment Questions).
Choose the right answer
1. If a > b, then is
b
a
A) Positive
B) Negative
C) Zero
2. Choose the right answer
a
b
b
a
is equal to
A) b
a
2
B) 0
C) 2 (a – b)
D) 2 (b – a)
An Important Logical Symbol
In Mathematics, we use symbols instead of sentences. For example, “3 is
greater than 2” is written as 3 > 2. Throughout the test we used the symbol
(read as “implies”)
If x> 2, then 2x > 4 is written as (x > 2) (2x > 4).
Generally ‘If P, then Q” is written as
P Q. (P is given and Q is the conclusion)
Note: P Q is different from Q P. Q P is called the converse of P Q.
The distance function
If a and b are two real numbers. Then the distance between a and b is
defined as | a – b|. Refer Figure 5.2. Why we choose |a – b| as the distance
between a and b should be clear from figure 5.2. When b > a, then the
distance is b – a; when a > b, it is a – b. Both a – b and b – a are equal to
|a – b|. So wherever a and b are on the real line, the distance is |a – b|.
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Fig. 5.2: Distance function
The distance function satisfies the following properties.
1. b
a
0
b
a
2. a
b
b
a
3. b
c
c
a
b
a
5.3 The Concept of Limit
In mathematics, the concept of a "limit" is used to describe the value that a
function or sequence "approaches" as the input or index approaches some
value. Limits are essential to calculus (and mathematical analysis in
general) and are used to define continuity, derivatives and integrals.
Function of a discrete variable and a continuous variable
The Concept of limit is associated with functions. A function from a set A to
a set B is a rule which assigns, to each element of A a unique (one and only
one) element of B. An example of a function is f(x) = 2x, a function which
associates with every number the number twice as large. Thus 5 is
associated with 10, and this is written f(5) = 10.
There are two types of functions. Discrete function and continuous
function.
Definition A function of a discrete variable or a discrete function is a
function from N or a subset of N to the set R of all real numbers.
a b
b – a
b a
a – b
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The second type of functions refer to functions from R to R. It is called a
function of a continuous variable.
Definition: A function of a continuous real variable or simply a function of a
real variable is a function from R to R.
Functions of a discrete variable
We have defined a function of a discrete variable as a function from
N {1, 2, 3, …} or subset of N to the set R of real numbers. A convenient way
to representing this function is by listing the images of 1, 2, 3, etc. If f
denotes the function then the list.
f(1),f(2),f(3), ….. ………………… (*)
represents the function f usually f(1), f(2),… are written as a1, a2,… etc.
The list given in (*) is called a sequence.
In a sequence the order of the elements appearing in it is important. A
common example of a sequence is a queue you see in a reservation
counter. Then a1 is the person standing in front of the counter getting his
reservation done. a2 is the person behind a1 etc. the order of persons in the
queue is important. You won’t certainly be happy if the order of the persons
in the queue is changed.
The limit of a sequence
From the above discussion, two points should be clear to you.
1. A sequence is an arrangement of real numbers as the first element,
second element etc.
2. A sequence represents a function of a discrete variable.
We denote a sequence by (an) and an denotes the nth term.
Assume that you have a string of length 1 cm. Denote it by a1. Cut the string
into two halves and throw away one half. Denote the remaining half by
a2.Then
2
1
a2 . Repeat the process indefinitely.
Then
3
4
2
3
2
1
a
,
2
1
a
etc. After (n + 1) repetitions, you are left with a
string of length .
2
1
a
n
1
n
Intuitively you feel that the string becomes
smaller and smaller and you are left with a string whose length is nearly
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zero in the long run. At the same time you realize that you will have “some
bit” of positive length at any time. Also you can make the string as small as
you please provided you repeat the process sufficient number of times. In
this case we say that “an tends to 0 as n tends to infinity” “an tends to 0”
means 0
an is as small as we please “n lends to infinity” means we
repeat the process sufficient number of times.
Now we are in a position to define the limit of a sequence (an)
Definition: Let (an) be a real sequence. Then (an) tends to a number a, if
given a positive number , (pronounced as epsilon), there exists a natural
number n0 such that
a
an for all 0
n
n ……………. (1.1)
In this case we write .
a
a
It
or
a
a n
a
n
n
We also say (an) converges to a.
Note: a
an is the numerical value of an – a. For example | 2 | = 2 and
| – 3 | = 3, and n0 is a “stage”.
n > n0 means after a certain stage,
a
an simply means that an comes
as close to a as we choose.
Example: Show that .
n
1
a
where
0
a n
n
Solution: .
n
1
0
a
So
.
n
1
0
n
1
0
a n
n
Let be a given positive number.
n
1
0
an when
1
n .
Let n0 be the smallest natural number .
1
(For example, if ,
7
.
147
1
take n0 = 148). Then .
n
1
0
If n > n0, then .
n
1
n
1
0
Hence
0
an when .
n
n 0
This proves (1.1) with 0 in place of a.
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Hence .
0
)
a
( n
Example: Show that 0
an where n
n
2
1
a .
Solution: As in Example, .
2
1
0
a n
n
For ,
2
1
n
we require
1
log
n
or
1
2 2
n
By choosing n0 to be the smallest natural number greater than
1
log2 , we
see that (1.1) is satisfied for n0.
Hence .
0
2
1
n
You can see that several similar sequences tend to 0. Some of them are
,.
log
1
........
,
4
1
,
3
1
.......
,
n
1
,
n
1
,
n
1
n
n
n
4
3
2
…… (1.2)
Subsequence: A subsequence is a sequence that can be derived from the
given sequence by deleting some elements from the sequence without
changing the order of the remaining elements. For example, ABD is a
subsequence of ABCDEF.Formally, suppose that X is a set and that (an)n∈N
is a sequence in X. Then, a subsequence of (an) is a sequence of the form
where (nr) is a strictly increasing sequence in the index set N.
Example:
1. Consider the sequence,
Then is a subsequence.
2. {1,1,1,1,....} is subsequence of {1,-1,1,-1,1,-1,....}. Algebra of limits of
sequences
If (an) and (bn) are two sequences, then we can get a new sequence by
“adding them”. Define cn = an + bn. Then (cn) is a sequence and we can write
(cn) = (an + bn). We can also subtract one sequence (bn) from another
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sequence (an), multiply two sequences, divide two sequences, etc. We can
also multiply a sequence (an) by a constant k.
Let us answer the following questions.
1. What happens to the limit of sum of two sequences ?
2. What happens to the limit of difference, product, division of two
sequences ?
We summarize the results as a theorem.
Theorem(): It (an) and (bn) and two sequences converging to a and b
respectively, then
a) (an + bn) → a + b
b) (an – bn) → a – b
c) (kan) → ka
d) (anbn) → ab
e)
b
a
b
a
n
n
providedbn 0 for all n and b 0.
Proof: We prove only a) Let > 0 be a positive number.
As (an) → a, we can apply (1.1) by taking
2
in place of . Thus we get a
natural number n0 such that
2
a
an
for all 0
n
n ………….. (1.3)
Similarly, using the convergence of (bn), we can get n1 such that
2
b
bn
for all 1
n
n …………….. (1.4.
Let m = maximum of n0 and n1. Then (1.3) and (1.4) are simultaneously true
for n > m.
Thus,
2
b
b
,
2
a
a n
n
for all m
n ………………… (1.5)
From (1.5) we get
2
2
b
b
a
a
b
b
a
a
b
a
b
a n
n
n
n
n
n
forn > m.
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Thus (1.1) holds good for the sequence (an + bn) in place of (an) and a + b
(in place of a)
Hence (an + bn) → a + b
Note: The choice of m may puzzle you. When
2
a
an
for all n >1000,
then certainly
2
a
an
for all n >1001, 1002 etc. So
2
a
an
for all
n > m, m being greater than 1000.
Remark: The other subdivisions can be proved similarly. As you are more
interested in applications you need not get tied down by the technical details
of the proof.Convergence of Sequence: A sequence is said to be
convergent if it has unique limit point.
Example:
1. (1/n) has 0 as its only limit point so it is convergent.
2. (1,-1,1,-1,1,….) has 1 and -1 as its limit pointAs the limit of its
subsequence (1,1,1,…) is 1 and limit of subsequence(-1,-1,-1,….) is
-1. So this sequence is not convergent. Such sequences are called
oscillatory sequences.
Ailter Proof:
We can write the sequence as (an) where
even
is
n
when
1
odd
is
n
when
1
an
Suppose (an) → a for some real number a. The number has to satisfy one
and only one of the following conditions: a < –1, –1 < a < 1, a > 1.
(See Fig. 5.3 representing these cases).
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Fig. 5.3: Illustration of W.E.
In case 2
,
1
a
an if n is odd. So we cannot prove (1.1) in this case
(for > 2).
In case 2
a
a
,
3 n
if n is even and we cannot prove (1.1) for > 2.
In case 2, if a is closer to 1, then 2
a
an for even n. If a is close to –1,
then 2
a
an for odd n. If a = 1, then 2
a
an for even n. So (1.1)
cannot hold good for > 2 or = 2. So the given sequence is not convergent.
Worked Examples
W.E.: Show that a constant sequence is convergent (A sequence (an) is a
constant sequence if an = k for all n).
Solution Since an = k for all n.Consider k
an .
0
k
k
k
an
As 0 <, for all n >1, that is, n0 = 1and l for every positive number > 0 ,the
sequence converges. So a constant sequence converges to its constants
value.
W.E.: Find the nth term of the sequence .....
..........
,
5
7
,
4
6
,
3
5
,
2
,
3 and find
its limit, if it exists.
Solution: To discover a pattern in the terms of the sequence start from the
third term.
3
3
2
3
5
a3
4
4
2
4
6
a4
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5
5
2
5
7
a5
6
6
2
6
8
a6
So a positive choice is
,
n
n
2
an
when an is written in this way
,
3
1
1
2
a1
,
3
5
3
3
2
a
,
2
2
2
a 3
2
etc.
To find the limit of the sequence (if it exists), with 1
n
2
n
n
n
2
an
Taking
n
1
bn and cn = 1, we get an = 2bn + cn
As 0
n
1
bn
and ,
1
1
n
c
1
1
0
2
an
, as n tends to infinity Hence the given sequence
converges to 1.
W.E.: Evaluate 2
2
n n
4
n
3
2
n
n
2
It
Solution As we know that 0
n
1
and ,
0
n
1
2
we try to write the nth
term of the given sequence in terms of .
n
1
and
n
1
2
2
2
n
n
4
n
3
2
n
n
2
a
2
2
2
2
n
n
4
n
3
2
n
n
n
2
4
n
3
n
2
1
n
1
n
2
2
2
As 0
n
1
,
n
1
2
and (1) → 1,
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1
1
0
0
2
1
It
n
1
It
n
1
It
2
1
n
1
n
2
It 2
2
n
Similarly,
4
4
0
3
0
2
4
n
3
n
1
It 2
n
Hence
4
1
n
4
n
3
2
n
n
2
It 2
2
n
S.A.Q. 3: Which of the following sets can be arranged as a sequence?
a) The passengers in a 3 ties coach
b) The people attending a meeting in a beach
c) The people living in Karnataka
d) The students of M.Sc. Biotechnology in a college
The Limit of a Function of a Real Variable
You are now familiar with natural numbers and real numbers. The natural
numbers appear as “discrete” points along the real line and we are able to
fix some element say 1 as the first natural number, 2 as the second natural
number etc. So the natural numbers appear as the terms of a sequence. But
it is not possible to arrange the real numbers as a sequence. If a real
number a is the nth
element and b is the (n+1)th
element, where will you
place
2
b
a
. It appears between the nth
element and the (n+1)th
element. If
you take
2
b
a
as the (n+1)th
element, where will you place
?
2
b
a
a
2
1
So you feel initiatively that real numbers can not be arranged as a sequence.
When we consider numbers between a and b. We consider points lying
between the points representing the numbers a and b. The numbers lying
between two numbers a and b form an “interval”. So “interval” on the real line
is the basic concept. Usually we define a function of a real variable on an
interval. We define various types of “intervals” as follows (refer to Table 5.1)
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Table 5.1: Intervals
Set notation Interval Graphical representation
}
b
x
a
|
R
x
{
(a, b)
}
b
x
a
|
R
x
{
[a, b]
}
b
x
a
|
R
x
{
[a, b)
}
b
x
a
|
R
x
{
(a, b]
}
x
a
|
R
x
{
[a, )
}
x
a
|
R
x
{
(a, )
}
|
{ a
x
R
x
(–, a]
}
a
x
|
R
x
{
(–, a)
R (–,)
Note: [Here (represents inclusion of all numbers > a.[ represents the
inclusion of all numbers > a.
is not a number. It simply represents the
inclusion of all “large” –ve numbers.
represents the inclusion of all
“large” positive real numbers.
(a, b) and [a, b] are called open interval and closed interval.]
So it is natural to represent R as the interval
, S.A.Q.4
a) Find all natural numbers in the intervals [3, ), (3, ), (– , 3) and
(– , 3]
b) Find all integers lying in the intervals given above
c) Find all numbers in [2, 2], (2, 2)
a b
a b
a b
a b
a
a
a
a
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Example; Represent
2
3
x
/
R
x
as an interval
Solution: 2
3
x
represents two inequalities x – 3 < 2, – (x – 3) < 2.
Whenx – 3 < 2, x < 5
When – (x – 3) < 2, x – 3 > – 2, or x > 3 – 2 = 1
Hence the given interval is (1, 5)
We can also arrive at this interval geometrically (see Fig 5.4)
Fig. 5.4
S.A.Q. 5: Represent the sets
2
3
x
/
R
x
,
2
3
x
/
R
x
,
2
3
x
/
R
x
as intervals.
Now we have enough background to define the limit of a function f of a real
variable x. In the case of functions of a discrete variable or sequence, we
defined )
n
(
f
It
or
a
It
n
n
n
. This limit represented the long term behaviour
of f. In the case of a functions of a real variables, we can discuss the
behaviour of f(x), when the variable x comes close to a real number a. In
other ways we will be defining ).
x
(
f
It
a
x We want to write the statement “x
comes close to a” rigorously. The geometric representation of real numbers
can be used for this purpose. When do you say that your house is near your
college? When the distance between your house and your college is small.
In the same way, we can say that “x is close to a “when | x – a| is small. If
“smallness” is defined by a distance of say 0.1, then x is close to a if
|x – a| < 0.1. Of course the measure of “smallness” is relative. For a person
living in Mangalore, Manipal is not near Mangalore. For a person living in
US, Mangalore and Manipal are near to each other. So “smallness” is
decided by the choice of a positive number (This was done in defining the
limit of a sequence also)
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Before giving a rigorous definition of limit, let us consider two examples.
Consider f(x) = 1 + x. Let us try to see what happens when x is close to 1.
We evaluate f(x), when x = 0.9, 0.99, 0.999, 1.1, 1.01, 1.001
f(0.9) = 1.9 f(0.99) = 1.99 f(0.999) = 1.999
f(1.1) = 2.1 f(1.01) = 2.01 f(1.001) = 2.001
Note all these values are near the value 2.
Consider another function .
1
x
,
1
x
1
x
x
g
2
(Why don’t we define g(1) ? If we put x = 1 in ,
1
x
1
x2
then we get
0
0
which
is not defined).
As in the case of f(x), we compute some function values.
99
.
1
99
.
0
g
9
.
1
1
9
.
0
1
9
.
0
9
.
0
g
2
g(0.999) = 1.999 g(1.1) = 2.1
g(1.01) = 2.01 g(1.001) = 2.001
Note: These values are close to 2. Hence we can say that x close to
1 both f(x) and g(x) are closed to 2 and we can take 2 as
x
f
It
1
x
or
x
g
It
1
x
.
Now let us formulate a rigorous definition of
x
f
It
a
x
Definition: Let f be a function of a real variable. Then I
x
f
lt
a
x
if given
a positive number , there exists a positive number such that
I
x
f
a
x ,
0 …………. (1.6)
Let us analyze the definition.
We have two choice of positive numbers (and ) and two conditions
I
x
f
and
a
x
0 Given any positive number , there exists
a positive number such that the condition P: "
a
x
0
"
Implies the
condition Q : "
"
I
x
f
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The choice of depends on the given number . The function f need not be
defined at x = a.
The condition P says that x is close to a.
The condition Q says that f(x) is close to I
We can also express the definition geometrically.
Given > 0, there exists > 0 such that
I
I
x
f
a
a
a
a
x ,
,
,
In figure 5.5, the point (a, I) is shown as 0, meaning that the functional value
of f at x = a is not known or defined.
Fig. 5.5: Definition of limit of function
The images of points in
a
,
a
)
a
,
a
( under the function f is a subset
of the interval (I – , I + ) along the vertical axis.
W.E.: Evaluate
6
x
2
It
3
x
Solution: Here f(x) = 2x – 6
Choose any > 0. We have to choose a such that (1.6) is satisfied. We
have to guess the value of I. When x comes close to 3,2x – 6 should come
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close to 2(3) – 6 = 0. Take I = 0. Then
0
6
x
2
0
)
x
(
f
3
x
2
2
3
x
So, choose ,
2
then
0
)
x
(
f
2
3
x
0
Hence, 0
6
x
2
It
3
x
Note: f(3) = 2(3) – 6 = 0. In this case the function f is defined at x = 3 and
f(3) coincides with
x
f
It
3
x
.
W.E.: Evaluate
1
x
2
It 2
0
x
Solution: Let 0
. As in the previous problem, we can guess the value of I
it is 1
1
)
0
(
2 2
1
1
x
2
1
x
f 2
2
x
2
2
x
Hence for a given 0
, the corresponding is chosen as
2
and condition
(1.6) holds good. Hence 1
1
x
2
It 2
0
x
W.E.: Evaluate x
It
0
x
Solution: Proceed as in the previous example. In this problem 2
and
0
x
It
0
x
18. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 115
General Remark While evaluating ,
x
f
It
a
x
if f(a) is defined or it is not of
the form ,
0
0
and f(x) is defined by a single expression, it will turn out that
.
a
f
x
f
It
a
x
If f(a) is not defined or f is given by two different expressions.
Then we have to guess the value of the limit and prove condition (1.6).
In some problems, f(x) may be given as quotient of two expressions but it
may reduce to an easier function on simplification. In such cases the
problem will reduce to an easier one.
W.E.: Evaluate
2
x
2
x
3
x
2
It
2
2
x
Solution: When x = 2, f(x) is of the form .
0
0
So we try to see when x – 2 is a
factor of 2x2
– 3x – 2.
2x2
– 3x – 2 = 2x2
– 4x + x – 2 = 2x (x – 2) + (x – 2) =
(2x +1) (x – 2).
When x 2, x does not assume the value 2. So,
2
x
2
x
1
x
2
2
x
2
x
3
x
2 2
= 2x + 1 on canceling x – 2, since x – 2 0.
So the given limit reduces to
5
1
x
2
It
2
x
Algebra of Limits of Functions
It is not necessary that we use the – definition for every problem. We
study important properties of limits of functions as a theorem. We can
evaluate limits using this theorem.
Theorem I:
a) a
x
It
a
x
b) k
k
It
a
x
, where k is a constant
c) 2
2
a
x
a
x
It
19. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 116
d) 3
3
a
x
a
x
It
e) n
n
a
x
a
x
It
f) a
x
It
a
x
when a > 0
Theorem 2: Let k be a constant, f and g functions having limit at a and n a
positive integer. Then the following hold good.
a)
x
f
It
k
x
kf
It
a
x
a
x
b)
x
g
It
x
f
It
x
g
x
f
It
a
x
a
x
a
x
c)
x
g
It
x
f
It
x
g
x
f
It
a
x
a
x
a
x
d)
x
g
It
.
x
f
It
x
g
.
x
f
It
a
x
a
x
a
x
e)
,
x
g
It
x
f
It
x
g
x
f
It
a
x
a
x
a
x
provided 0
x
g
It
a
x
f)
n
a
x
n
a
x
x
f
It
x
f
It
g
x
f
It
x
f
It
a
x
a
x
provided
x
f
lt
a
x
is positive
You need not prove these results. It is enough if you clearly understand the
theorems and apply them for evaluating limits.
Self Assessment Questions
SAQ 6: Evaluate the following limits
a)
n
2
1
n
2
It
n
b)
1
n
n
It
n
c)
100
It
n
d) 2
2
n n
2
n
3
5
7
n
4
n
3
It
e) 2
2
n n
n
1
n
n
It
f)
3
3
n n
1
n
1
It
g) 2
2
n n
n
1
n
2
It
h) 2
2
n n
n
9
n
3
2
It
20. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 117
SAQ 7: Evaluation the following limits
a)
2
1
x
x
x
1
It
b) 2
1
x x
x
1
1
It
c) 2
0
x
x
4
x
3
2
It
d)
2
x
4
x
It
2
2
x
e)
1
x
1
x
It
3
1
x
f) 2
2
1
x x
x
1
x
3
x
2
It
g)
2
x
2
x
3
x
It
2
2
x
h)
1
x
1
x
It 2
1
x
S.A.Q.8: If ,
m
x
g
It
and
I
x
f
It
a
x
a
x
evaluate the following
a)
x
g
3
x
f
2
It
a
x
b)
x
g
x
f
x
g
x
f
It
a
x
c)
2
2
a
x
x
g
x
f
It
d)
2
a
x
x
g
1
x
f
It
e)
x
g
4
x
g
2
x
f
It
a
x
if m is positive f)
x
f
2
x
g
It
a
x
if I, m > 0.
5.4 Concept of Continuity
In mathematics and sciences, we use the word “continuous” to describe a
process that goes on without abrupt changes. For example, the growth of a
plant, the water level in a tank and the speed of a moving car in a four-base
highway are exhibiting continuous behaviour.
Before defining continuous functions, let us look at the graphs of three
functions.
Fig. 5.6: Two discontinuous functions and a continuous function
The first graph has a break at x = a in the second graph also there is a
break at x = a. If you ignore the point corresponding to x = a, there is no
21. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 118
break for the break occurs at x = a the third function has no break. So it
should be intuitively clear to you that the first two functions are not
continuous while the third function is continuous.
Let us formulate a rigorous definition of continuity.
Definition: Let f be a function of a real variable defined in an open interval
containing a. Then f is continuous at a if
a
f
x
f
It
a
x
Note: In order to define continuity at a, we need three conditions.
1)
x
f
It
a
x
exists
2) f(a) is defined
3) )
a
(
f
x
f
It
a
x
Even if any one of them fails, then the function f is not continuous at a.
Now look at Fig. 5.6. The first function, say f, has no limit at a, i.e.,
x
f
It
a
x
does not exist. For the second function,
x
f
It
a
x
exists but
.
a
f
x
f
It
a
x
The third function is continuous.
Example: Define f as follows:
2
x
if
3
2
x
if
2
x
4
x
x
f
2
Is f continuous at 2?
Solution:
2
x
2
x
2
x
It
2
x
4
x
It
2
x
2
2
x
2
x
It
2
x
= 4
So
x
f
It
x 2
exists.
22. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 119
But f(2) = 3 4
x
f
It
2
x
Hence the function f is not continuous 2.
Example: Define f as follows.
2
x
if
4
2
x
if
2
x
4
x
x
f
2
Is f continuous at 2?
Solution: From above example, we have .
4
x
f
It
2
x
As f(2) = 4, f is
continuous at 2.
Sometimes function may be defined by two different expressions. In such
cases the following method of proving continuity will be useful. For that we
need the concept of left limit and right limit.
Definition: Let a function f be defined in an interval (b, a) where b < a).
Then I
x
f
lt
a
x
(called the left limit) if for given > 0, there exists > 0
such that
I
x
f
a
a
x ,
………… (1.7).
Definition; Let a function f be defined in an interval (a, b), where b > a.
Then l
x
f
lt
a
x
called the right limit if given > 0, there exists > 0 such
that
I
x
f
a
,
a
x ………….. (1.8)
Note: We can prove that (1.7) and (1.8) implies (1.6). Hence if the left and
right limits exist and are equal then
)
x
(
f
It
)
x
(
f
It
x
f
It
a
x
a
x
a
x
When a function is defined by two different expressions, we have to
evaluate the left and right limits.
I
x
f
It
a
x
if both the left and right limits exist and are equal.
23. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 120
Worked Examples:
W.E.: Test whether f is continuous at x = 3 where f is defined by
3
x
if
2
3
x
if
4
x
3
x
f
Solution: As f is defined by two expressions one for ]
3
,
( and another
for (3, ), we evaluate the left and right limits.
5
4
3
3
4
3
3
3
x
lt
x
f
lt
x
x
2
2
3
3
x
x
lt
x
f
lt
As the left and right limits are not equal,
x
f
It
3
x
does not exist. Hence f is
not continuous at 3.
When a function f is continuous at every point of an interval (a, b) we say
that the function is continuous on (a, b). In particular, if a function is
continuous at every real number. Then we say that a function is continuous
on R.
Using Theorem 1, we can prove that the functions, k (a constant), x, x2
,
……..xn
, where n > 2 are continuous on R.
The function x is continuous on (0, )
The function
x
1
is continuous on
,
0
0
,
Using Theorem 2, we can prove the following theorem.
Theorem:
a) If f and g are continuous at a, then
i) f(x) + g(x) ii) f(x) – g(x) iii) f(x) g(x) are continuous at a
b) If f and g are continuous at a and g(a) 0, then
g
f
is continuous at a
c) If f is continuous at a, f(x) > 0 for x in an open interval containing a,
x
f is continuous at a.
24. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 121
Worked Examples
W.E.: Test the continuity of the function f at all real points where f is defined
by
3
x
for
1
x
2
3
x
for
x
x
f
2
Solution: If a < 3, then f(x) is defined by the expression x2
in an open
interval containing a. So f is continuous for all a < 3.
So it remains to test continuity only at 3.
9
3
x
It
x
f
It 2
2
3
x
3
x
7
1
3
2
1
x
2
It
x
f
It
3
x
3
x
As
x
f
It
,
x
f
It
x
f
It
3
x
3
x
3
x
does not exist. So f is not continuous at 3.
Thus f is continuous at all real points except 3.
W.E.: Test the continuity of the function f where f is defined by
2
x
if
7
2
x
if
|
2
x
|
2
x
x
f
Solution: When x < 2, |x – 2| is negative. So |x – 2| = – (x – 2)
1
2
x
2
x
x
f
When x > 2, |x – 2| is positive. So |x – 2| = x – 2
1
2
x
2
x
x
f
f(2) = 7. Thus
2
1
2
7
2
1
x
if
x
if
x
if
x
f
As in the previous worked example, f is continuous for all a < 2 and all a > 2.
1
1
It
x
f
It
2
x
2
x
1
1
It
x
f
It
2
x
2
x
As
x
f
It
,
x
f
It
x
f
It
2
x
2
x
2
x
does not exist. So f is continuous at all
points except 2.
25. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 122
Note. In some cases
x
f
It
a
x
may not exist. You may ask a question: how
to establish that
x
f
It
a
x
does not exist. One simple case is where the left
and right limits exist but are not equal. We have examples for this case. The
worst case is when neither of the two limits exist. For proving the non-
existence of limits, we use the following theorem.
Theorem: A function f is continuous at a if and only if the following
conditions holds good.
a
f
x
f
a
x n
n
To prove the non existence of the limit it is enough to construct a sequence
(xn) converging to a such that f(xn) does not converge to f(a).
W.E. : Show that the function f defined by
0
x
if
0
0
x
if
x
1
x
f is not continuous at 0.
Solution: Let .
n
1
xn Then 0
xn (obvious) f(xn) = n and so f(xn) does
not converge to 0 since f(xn) indefinitely increases and so cannot approach
0.
Self Assessment Questions
S.A. Q. 9: Verify whether the following functions f is continuous at a
a)
1
a
1
x
if
5
1
x
if
3
x
2
x
f
b)
1
a
1
x
if
9
1
x
if
5
x
4
x
f
c)
2
a
2
x
if
x
1
2
a
2
x
if
x
1
x
f 2
d) 3
3
3
1
3
9
2
a
x
if
x
if
x
x
x
f
26. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 123
e) 3
3
6
3
3
9
2
a
x
if
x
if
x
x
x
f
f) 2
a
3
x
if
7
3
x
if
3
x
9
x
x
f
2
g) 3
a
3
x
if
6
3
x
if
9
x
3
x
x
f 2
S.A.Q. 10: Show that the function f defined by
1
x
if
0
1
x
if
1
x
1
x
f is not continuous at 1.
S.A.Q. 11: Show that the following functions are continuous on R.
a)
1
x
if
1
x
if
2
x
2
5
x
6
x
x
f 2
2
b)
1
x
if
4
1
x
if
x
x
x
1
x
f
3
2
c)
1
x
if
x
4
1
x
if
x
4
1
x
f 2
3
d)
1
x
if
1
x
1
x
if
2
x
3
x
x
f 3
2
5.5 Summary
In this unit, we studied the basics of real number system then the concept of
limit was discussed which was further extended to the concept of continuity.
All definitions and properties of the above mentioned concepts is given very
clearly with sufficient number of examples wherever necessary.
27. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 124
5.6 Terminal Questions
1. Find all natural numbers in the following intervals
a) [4, 9) b) (4, 9] c) (4, 9) d) [f, g]
e) (4, ) f) (9, ) g) [9, ) h) (4, ) (9, )
2. Find n
n
a
lt
when
a) n
n
2
3
a b) 1
3
2
a n
n
c) n
n
n
3
2
2
3
a
d)
!
n
1
an e)
2
n
1
an
3. Show n
n
a
lt
does not exist when
a) an = n b) an = 2n
c) an = n!
4. Evaluate n
n
a
lt
when
a)
7
n
4
2
n
3
an
b)
n
1
n
an
c)
n
n
3
2
an
d) 2
2
n
n
7
n
4
3
n
n
2
a
e) 2
2
n
n
n
1
n
n
1
a
f) 2
2
n
n
n
n
1
a
g)
n
2
3
n
2
an
h) 2
2
n
n
n
2
3
n
3
n
2
1
a
i)
3
n
2
4
n
1
n
3
n
a 2
2
n
j)
2
n
n
n
a 3
2
n
k) n
1
n
an
5. Evaluate n
n
a
lt
when f(x) is equal to
a) 3
x
2 b) 4
3
x
x c) 5
4
3
x
x
x
2
d)
2
x
5
4
x
3
2
e) 2
x
5
4
x
3
2
f)
4
9
x2
6. Evaluate the following limits
a)
1
x
2
It 2
1
x
b)
x
2
3
1
x
2
It
1
x
c) 2
1
x x
5
x
7
4
It
d) 3
x
3
It
3
x
e)
3
x
2
1
It
1
x
f)
2
3
1
2
2
2
3
x
x
x
x
It
x
28. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 125
g)
1
x
1
x
It 2
3
1
x
h)
1
x
2
x
x
It 2
2
1
x
7. If I
x
f
It
a
n
and ,
m
x
g
It
a
n
evaluate
a)
x
g
x
f
It
a
n
b)
x
g
x
f
2
x
g
2
x
f
It
a
n
c)
x
g
3
x
f
2
It
a
n
(when I, m, > 0)
d)
2
a
n x
g
x
g
x
f
It
whenm > 0
8. Show that the following functions are continuous at a
a) 2
2
x
x
1
x
f
for all x in R, a = 0
b) 2
x
1
1
x
f
for all x in R a = 0
c) 2
x
2
1
x
3
2
x
f
for all x in R a = 0
9. Show that the following functions are continuous at a
a) 4
a
4
x
if
8
4
x
if
4
x
16
x
x
f
2
b) 1
a
1
x
if
x
2
x
3
4
1
x
if
x
4
x
3
2
x
f 2
2
c)
1
a
1
x
if
x
1
1
x
if
x
1
x
f 3
2
10. Show that the following functions are not continuous at a
a)
2
a
2
x
if
7
2
x
if
2
x
8
x
2
x
f
2
b)
1
a
1
x
if
1
1
x
if
x
1
x
1
x
f
2
c)
1
a
1
x
if
x
4
x
3
1
x
if
x
4
x
3
1
x
f 2
2
29. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 126
d)
1
a
1
x
if
1
1
x
if
1
x
f
e)
0
a
0
x
if
1
0
x
if
0
0
x
if
1
x
f
5.7 Answers
Self Assessment Questions
1. A
2. A
3. a) and d) can be arranged as a sequence according to chart and
attendance register
4. (a) i) {3,4,5,…}, ii) {4,5,6,…}, iii) {1,2}, iv) {1,2,3}
b) (i) {3, 4, 5, …..} (ii){4, 5, 6, ……},iii){……. –3, –2, –1, 0, 1, 2,},
iv) {…. –3, –2, –1, 0, 1, 2, 3}
(C ) i) , ii)
5. [1, 5], (–, 1) (5, ), (–, 1] [5, )
6. a) 1 b) 1 c) 100 d)
2
3
e) 1 f) –1 g) 2 h) 9
7. a) 3 b)
3
1
c) 2 d) 4
e) 3 (Hint: x3
– 1 = (x – 1) (x2
+ x + 1)) f)
3
5
g) 1
h)
2
1
8. a) 2I + 3m b) I2
– m2
c) 2
2
m
I d)
2
m
1
1
e)
m
4
m
2
I
f) l
m 2
9. a), b), c) continuous d) discontinuous, e),f) continuous,
g) not continuous
30. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 127
10. Take
n
1
1
xn
,
0
n
1
1
but f(xn) = n, f(xn) does not converge. So
f is not continuous at 1.
11. a) For a < 1, f(x) = x2
– 6x + 5. So f is continuous for
a < 1. As f(x) = 2x2
– 2, f is continuous for a > 1.
.
0
2
2
x
f
It
,
0
5
6
1
x
f
It
1
x
1
x
Hence .
0
x
f
It
1
x
Also f(1) = 1 – 6 + 5 = 0
b), c), d) Similar.
Terminal Questions
1. a) {4, 5, 6, 7, 8} b) {5, 6, 7, 8, 9} c) {5, 6, 7, 8}
d) {f,g}
e) { 5, 6, …..} f) { 11, 12,…..} g) {9, 10, 11, …..}
h) { 10, 11, ….. }
2. a) 0 b) 1 c) 0 d) 0 e) 0
3. In all the case, As n increases an increases. So k
an cannot be
made less than a fixed number . So n
n
a
lt
does not exists.
4. a)
4
3
b) 1 c) 3 d)
7
1
e) – 1 f) 1 g)
2
1
h) 3
i) Write an as
2
1
1
2
1
1
.
4
1
1
1
.
3
2
3
1
2
2
n
n
a
lt
n
n
n
n
j) 0
0
1
0
0
1
0
0
.
2
1
1
1
3
2
n
n
n a
lt
n
n
n
a
k)
n
1
n
n
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an
31. Mathematics for IT Unit 5
Manipal University Jaipur B0947 Page No.: 128
n
n
n
n
n
n
n
n 1
1
1
.
1
1
1
1
1
1
= 0
1
1
0
a
It n
n
5. a) 2a3
b) a3
+ a4
c) 2a3
– a4
+ a5
d) (2 + 3a) (4 + 5a2
)
e) 2
a
5
4
a
3
2
f)
4
9
2
a
6. a) 2(–1)2
– 1 = 1 b) (2 + 1) (3 – 2) = 3 c)
6
11
d) 6
e)
5
1
f)
8
2
9
9
1
6
9
g) 1
h)
.
1
x
2
x
1
x
1
x
2
x
1
x
1
x
2
x
x
2
2
Hence answer is .
2
3
7. a) Im b) (l + 2m) (2I – m) c) m
3
I
2 d)
2
m
m
I
8. a)
8
x
f
It
2
x
2
x
2
x
2
2
x
8
x
2
2
x
2
f(2) = 7. So f is not
continuous at x=2.
b) .
2
x
1
It
x
1
x
1
It
x
f
It
1
x
2
1
x
1
x
But f(1) = 1
c) Left limit at 1 = 8; right limit at 1 = 2
d) The left and right limits are 1 and –1 respectively.
e) The left and right limits are –1 and 1.