Lecture 4 - Limit of Functions and One-Sided Limits | Calculus - Engineering Mathematics PDF Download

 Page 1


    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Limit of Functions and One-Sided Limits 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Page 2


    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Limit of Functions and One-Sided Limits 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
Table of Contents 
 Chapter: Limit of Functions and One-Sided Limits 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Limit of a Function (Imprecise Definition) 
? 4: Limit of a Function at a Point (?? ? Definition) 
o 4.1. Method to find the value of ? Algebraically 
? 5: Algebra of Limits 
? 6: One-Sided Limits 
o 6.1. Right Hand Limit at a Point 
o 6.2. Left Hand Limit at a Point 
? 7: Infinite Limits 
? Exercises 
? Summary 
? Reference 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Limit of Functions and One-Sided Limits 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
Table of Contents 
 Chapter: Limit of Functions and One-Sided Limits 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Limit of a Function (Imprecise Definition) 
? 4: Limit of a Function at a Point (?? ? Definition) 
o 4.1. Method to find the value of ? Algebraically 
? 5: Algebra of Limits 
? 6: One-Sided Limits 
o 6.1. Right Hand Limit at a Point 
o 6.2. Left Hand Limit at a Point 
? 7: Infinite Limits 
? Exercises 
? Summary 
? Reference 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Limit of a function (Imprecise Definition) 
? Limit of a function at a point (?? ? Definition) 
? Method to find the value of ? algebraically 
? Algebra of limits 
? One Sided Limits at a Point 
? Right Hand Limit at a Point 
? Left Hand Limit at a Point 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Limit of Functions and One-Sided Limits 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
Table of Contents 
 Chapter: Limit of Functions and One-Sided Limits 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Limit of a Function (Imprecise Definition) 
? 4: Limit of a Function at a Point (?? ? Definition) 
o 4.1. Method to find the value of ? Algebraically 
? 5: Algebra of Limits 
? 6: One-Sided Limits 
o 6.1. Right Hand Limit at a Point 
o 6.2. Left Hand Limit at a Point 
? 7: Infinite Limits 
? Exercises 
? Summary 
? Reference 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Limit of a function (Imprecise Definition) 
? Limit of a function at a point (?? ? Definition) 
? Method to find the value of ? algebraically 
? Algebra of limits 
? One Sided Limits at a Point 
? Right Hand Limit at a Point 
? Left Hand Limit at a Point 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
2. Introduction: 
 The central idea that distinguishes calculus from algebra and 
trigonometry is the concept of limit. The essence of the concept of limit 
for real-valued functions of a real variable is that: if ? is a real number, 
then 
0
lim ( )
xx
fx
?
? ? means that the value of f(x) can be made as close to ? as 
we wish by taking x sufficiently close to 
0
x (analyze the figure 1). 
 
Figure 1  
3. Limit of a Function (Imprecise Definition): 
Let f(x) be defined on an open interval I about 
0
x , except possibly at 
0
x 
itself. If f(x) get arbitrarily close to ? for all x sufficiently close to 
0
x , then 
the function f(x) approaches the limit ? as x approaches  
0
x and it is 
denoted by  
 
0
lim ( )
xx
fx
?
? ?  
and read as the limit of f(x) as x approaches 
0
x is ? . 
Value Addition: Note 
In the above definition of limit phrases like arbitrarily close and 
sufficiently close are imprecise because the meaning of these phrases 
depends on the context. To a biologist, close may mean within a few 
thousandths of an inch. To an astronomer studying distant galaxies close 
may mean within a few thousand light-years. 
 
Example 1: Check the behavior of the function 
Page 5


    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Limit of Functions and One-Sided Limits 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, A.N.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
Table of Contents 
 Chapter: Limit of Functions and One-Sided Limits 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Limit of a Function (Imprecise Definition) 
? 4: Limit of a Function at a Point (?? ? Definition) 
o 4.1. Method to find the value of ? Algebraically 
? 5: Algebra of Limits 
? 6: One-Sided Limits 
o 6.1. Right Hand Limit at a Point 
o 6.2. Left Hand Limit at a Point 
? 7: Infinite Limits 
? Exercises 
? Summary 
? Reference 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Limit of a function (Imprecise Definition) 
? Limit of a function at a point (?? ? Definition) 
? Method to find the value of ? algebraically 
? Algebra of limits 
? One Sided Limits at a Point 
? Right Hand Limit at a Point 
? Left Hand Limit at a Point 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
2. Introduction: 
 The central idea that distinguishes calculus from algebra and 
trigonometry is the concept of limit. The essence of the concept of limit 
for real-valued functions of a real variable is that: if ? is a real number, 
then 
0
lim ( )
xx
fx
?
? ? means that the value of f(x) can be made as close to ? as 
we wish by taking x sufficiently close to 
0
x (analyze the figure 1). 
 
Figure 1  
3. Limit of a Function (Imprecise Definition): 
Let f(x) be defined on an open interval I about 
0
x , except possibly at 
0
x 
itself. If f(x) get arbitrarily close to ? for all x sufficiently close to 
0
x , then 
the function f(x) approaches the limit ? as x approaches  
0
x and it is 
denoted by  
 
0
lim ( )
xx
fx
?
? ?  
and read as the limit of f(x) as x approaches 
0
x is ? . 
Value Addition: Note 
In the above definition of limit phrases like arbitrarily close and 
sufficiently close are imprecise because the meaning of these phrases 
depends on the context. To a biologist, close may mean within a few 
thousandths of an inch. To an astronomer studying distant galaxies close 
may mean within a few thousand light-years. 
 
Example 1: Check the behavior of the function 
    Limit of Functions and One-Sided Limits  
 
Institute of Lifelong Learning, University of Delhi                                                  
 
 
2
4
()
2
x
fx
x
?
?
?
,  near x = 2. 
Solution: Given that 
 
2
4
()
2
x
fx
x
?
?
?
 
for any 2 x ? , we can have 
 
( 2)( 2)
( ) 2 2
2
xx
f x x for x
x
??
? ? ? ?
?
. 
Thus, the graph of f(x) is the line 2 yx ?? with the point (1,  2) removed. 
 
Figure 2: Graph of 
2
4
()
2
x
fx
x
?
?
?
. 
The removed point (2, 4) is shown as a hole in the graph. Even though 
f(2) is not defined, it is clear that we can make the value of f(x) as close 
as we want to 4 by choosing x close enough to 2. 
Thus, f(x) approaches the limit 4 as x approaches to 2. 
Hence, 
 
2
22
4
lim ( ) lim 4
2
xx
x
fx
x
??
?
??
?
. 
Example 2: Discuss the behavior of function 
 
1, 0
()
2, 0
x
fx
x
? ?
?
?
?
?