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Page 1
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Limits and Continuity of Functions of several
Variables
Lesson Developer: Kapil Kumar
Department/College: Assistant Professor, Department of
Mathematics, A.R.S.D. College, University of Delhi
Page 2
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Limits and Continuity of Functions of several
Variables
Lesson Developer: Kapil Kumar
Department/College: Assistant Professor, Department of
Mathematics, A.R.S.D. College, University of Delhi
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
Chapter: Limits and Continuity of Functions of several Variables
? 1: Learning outcomes
? 2: Introduction
? 3: Functions of several variables
? 4: Limits of functions of several variables
o 4.1. Non-existence of limit
o 4.2. Determining the simultaneous limits by changing to
polar coordinates
? 5: Algebra of limits
? 6: Repeated limits or iterative limits
? 7: Two-path test for non-existence of a limit
? 8: Continuity at a point
o 8.1. ?? ? definition of continuity of a function at a point
? Exercises
? Summary
? References
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Functions of several variables
? Limits of functions of several variables
? Algebra of limits
? Repeated limits or iterative limits
? Two-path test for non-existence of a limit
? Continuity at a point
Page 3
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Limits and Continuity of Functions of several
Variables
Lesson Developer: Kapil Kumar
Department/College: Assistant Professor, Department of
Mathematics, A.R.S.D. College, University of Delhi
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
Chapter: Limits and Continuity of Functions of several Variables
? 1: Learning outcomes
? 2: Introduction
? 3: Functions of several variables
? 4: Limits of functions of several variables
o 4.1. Non-existence of limit
o 4.2. Determining the simultaneous limits by changing to
polar coordinates
? 5: Algebra of limits
? 6: Repeated limits or iterative limits
? 7: Two-path test for non-existence of a limit
? 8: Continuity at a point
o 8.1. ?? ? definition of continuity of a function at a point
? Exercises
? Summary
? References
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Functions of several variables
? Limits of functions of several variables
? Algebra of limits
? Repeated limits or iterative limits
? Two-path test for non-existence of a limit
? Continuity at a point
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
In studying a real world phenomenon and applications in geometry, applied
mathematics, engineering and natural science, a quantity being investigated
usually depends on two or more independent variables. Therefore we need
to extend the basic ideas of the calculus of functions of a single variable to
functions of several variables. In this lesson we will study the limits and
continuity for multivariable functions. Although the definitions of the limit of
a function of two or three variables is similar to the definition of the limit of a
function of a single variable but with a crucial difference.
3. Functions of Several Variables:
Real valued functions of several independent real variables are defined in the
same way as the real valued functions of single variable. The domains of the
real valued functions of several variables are the sets of ordered pairs
(triples, quadruples, n-tuples) of real numbers and the ranges are subsets of
real numbers.
For example: 1. Consider the function
2
V r h ? ? , here V denoted the volume
of he cylinder, r radius and h height of the cylinder. Here V depends on r
and h. Thus, r and h are called the independent variables and V is called
dependent variable.
2. The relation
22
1 Z x y ? ? ? , between x, y and z determines a value of z
corresponding to every pair of numbers x, y which are such that
22
1 xy ?? .
The region determined by the point (x, y) is called the domain of the point
(x, y).
3. The relation
22
xy
Ze
?
? determines a function of two variables (x, y); the
domain of the function being the whole plane i.e., the set of all the ordered
pairs of real numbers.
Definition 1: A variable Z is said to be a function of two variables x and y,
denoted by ( , ) Z f x y ? , if to each pair of values of x and y (over same domain
D) there corresponds a definite value of Z. Here x and y are called the
independent variables and Z is called the dependent variable.
Definition 2: Let D is a set of n-tupple of real numbers
12
( , , ..., )
n
x x x . A real-
valued function f on D is a rule that assign a unique real number
Page 4
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Limits and Continuity of Functions of several
Variables
Lesson Developer: Kapil Kumar
Department/College: Assistant Professor, Department of
Mathematics, A.R.S.D. College, University of Delhi
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
Chapter: Limits and Continuity of Functions of several Variables
? 1: Learning outcomes
? 2: Introduction
? 3: Functions of several variables
? 4: Limits of functions of several variables
o 4.1. Non-existence of limit
o 4.2. Determining the simultaneous limits by changing to
polar coordinates
? 5: Algebra of limits
? 6: Repeated limits or iterative limits
? 7: Two-path test for non-existence of a limit
? 8: Continuity at a point
o 8.1. ?? ? definition of continuity of a function at a point
? Exercises
? Summary
? References
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Functions of several variables
? Limits of functions of several variables
? Algebra of limits
? Repeated limits or iterative limits
? Two-path test for non-existence of a limit
? Continuity at a point
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
In studying a real world phenomenon and applications in geometry, applied
mathematics, engineering and natural science, a quantity being investigated
usually depends on two or more independent variables. Therefore we need
to extend the basic ideas of the calculus of functions of a single variable to
functions of several variables. In this lesson we will study the limits and
continuity for multivariable functions. Although the definitions of the limit of
a function of two or three variables is similar to the definition of the limit of a
function of a single variable but with a crucial difference.
3. Functions of Several Variables:
Real valued functions of several independent real variables are defined in the
same way as the real valued functions of single variable. The domains of the
real valued functions of several variables are the sets of ordered pairs
(triples, quadruples, n-tuples) of real numbers and the ranges are subsets of
real numbers.
For example: 1. Consider the function
2
V r h ? ? , here V denoted the volume
of he cylinder, r radius and h height of the cylinder. Here V depends on r
and h. Thus, r and h are called the independent variables and V is called
dependent variable.
2. The relation
22
1 Z x y ? ? ? , between x, y and z determines a value of z
corresponding to every pair of numbers x, y which are such that
22
1 xy ?? .
The region determined by the point (x, y) is called the domain of the point
(x, y).
3. The relation
22
xy
Ze
?
? determines a function of two variables (x, y); the
domain of the function being the whole plane i.e., the set of all the ordered
pairs of real numbers.
Definition 1: A variable Z is said to be a function of two variables x and y,
denoted by ( , ) Z f x y ? , if to each pair of values of x and y (over same domain
D) there corresponds a definite value of Z. Here x and y are called the
independent variables and Z is called the dependent variable.
Definition 2: Let D is a set of n-tupple of real numbers
12
( , , ..., )
n
x x x . A real-
valued function f on D is a rule that assign a unique real number
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 4
12
( , , ..., )
n
Z f x x x ?
To each element in D. The set D is called the domain and the set of z-values
taken on by f is the function’s range. The
12
, , ...,
n
x x x are called independent
variable and the Z is called a function of n independent variables.
4. Limits:
The definition of the limit of a function of two or three variables is
similar to the definition of the limit of a function of a single variable but with
a crucial difference.
A function ( , ) f x y is said to tend to a limit as a point ( , ) xy tends to the
point
00
( , ) xy if for every arbitrarily small positive number ? , there exists a
positive number 0 ? ? such that
( , ) f x y ? ?? whenever
0
0 xx ? ? ? ? ,
0
0 yy ? ? ? ?
Or ? ? ? ?
22
00
0 x x y y ? ? ? ? ? ?
Symbolically, the limit of the function ( , ) f x y at the point
00
( , ) xy is denoted
by
00
( , ) ( , )
lim ( , )
x y x y
f x y
?
?
Or
0
0
lim ( , )
xx
yy
f x y
?
?
?
Where is called the limit (the double limit or the simultaneous limit) of f
when ( , ) xy tends to
00
( , ) xy simultaneously.
Value Addition: Note
1. The definition of limit says that the distance between ( , ) and f x y
becomes arbitrarily small whenever the distance from ( , ) xy to
00
( , ) xy is
made sufficiently small (but not 0).
2. The simultaneous limit postulates that by whatever path the point is
approached, the function f attains the same limit.
3. In general the determination whether a simultaneous limit exists or not is
a difficult matter but very often a simple consideration enables us to show
that the limit does not exist.
Page 5
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 1
Lesson: Limits and Continuity of Functions of several
Variables
Lesson Developer: Kapil Kumar
Department/College: Assistant Professor, Department of
Mathematics, A.R.S.D. College, University of Delhi
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents
Chapter: Limits and Continuity of Functions of several Variables
? 1: Learning outcomes
? 2: Introduction
? 3: Functions of several variables
? 4: Limits of functions of several variables
o 4.1. Non-existence of limit
o 4.2. Determining the simultaneous limits by changing to
polar coordinates
? 5: Algebra of limits
? 6: Repeated limits or iterative limits
? 7: Two-path test for non-existence of a limit
? 8: Continuity at a point
o 8.1. ?? ? definition of continuity of a function at a point
? Exercises
? Summary
? References
1. Learning outcomes:
After studying this chapter you should be able to understand the
? Functions of several variables
? Limits of functions of several variables
? Algebra of limits
? Repeated limits or iterative limits
? Two-path test for non-existence of a limit
? Continuity at a point
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 3
2. Introduction:
In studying a real world phenomenon and applications in geometry, applied
mathematics, engineering and natural science, a quantity being investigated
usually depends on two or more independent variables. Therefore we need
to extend the basic ideas of the calculus of functions of a single variable to
functions of several variables. In this lesson we will study the limits and
continuity for multivariable functions. Although the definitions of the limit of
a function of two or three variables is similar to the definition of the limit of a
function of a single variable but with a crucial difference.
3. Functions of Several Variables:
Real valued functions of several independent real variables are defined in the
same way as the real valued functions of single variable. The domains of the
real valued functions of several variables are the sets of ordered pairs
(triples, quadruples, n-tuples) of real numbers and the ranges are subsets of
real numbers.
For example: 1. Consider the function
2
V r h ? ? , here V denoted the volume
of he cylinder, r radius and h height of the cylinder. Here V depends on r
and h. Thus, r and h are called the independent variables and V is called
dependent variable.
2. The relation
22
1 Z x y ? ? ? , between x, y and z determines a value of z
corresponding to every pair of numbers x, y which are such that
22
1 xy ?? .
The region determined by the point (x, y) is called the domain of the point
(x, y).
3. The relation
22
xy
Ze
?
? determines a function of two variables (x, y); the
domain of the function being the whole plane i.e., the set of all the ordered
pairs of real numbers.
Definition 1: A variable Z is said to be a function of two variables x and y,
denoted by ( , ) Z f x y ? , if to each pair of values of x and y (over same domain
D) there corresponds a definite value of Z. Here x and y are called the
independent variables and Z is called the dependent variable.
Definition 2: Let D is a set of n-tupple of real numbers
12
( , , ..., )
n
x x x . A real-
valued function f on D is a rule that assign a unique real number
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 4
12
( , , ..., )
n
Z f x x x ?
To each element in D. The set D is called the domain and the set of z-values
taken on by f is the function’s range. The
12
, , ...,
n
x x x are called independent
variable and the Z is called a function of n independent variables.
4. Limits:
The definition of the limit of a function of two or three variables is
similar to the definition of the limit of a function of a single variable but with
a crucial difference.
A function ( , ) f x y is said to tend to a limit as a point ( , ) xy tends to the
point
00
( , ) xy if for every arbitrarily small positive number ? , there exists a
positive number 0 ? ? such that
( , ) f x y ? ?? whenever
0
0 xx ? ? ? ? ,
0
0 yy ? ? ? ?
Or ? ? ? ?
22
00
0 x x y y ? ? ? ? ? ?
Symbolically, the limit of the function ( , ) f x y at the point
00
( , ) xy is denoted
by
00
( , ) ( , )
lim ( , )
x y x y
f x y
?
?
Or
0
0
lim ( , )
xx
yy
f x y
?
?
?
Where is called the limit (the double limit or the simultaneous limit) of f
when ( , ) xy tends to
00
( , ) xy simultaneously.
Value Addition: Note
1. The definition of limit says that the distance between ( , ) and f x y
becomes arbitrarily small whenever the distance from ( , ) xy to
00
( , ) xy is
made sufficiently small (but not 0).
2. The simultaneous limit postulates that by whatever path the point is
approached, the function f attains the same limit.
3. In general the determination whether a simultaneous limit exists or not is
a difficult matter but very often a simple consideration enables us to show
that the limit does not exist.
Limits and Continuity of Functions of several Variables
Institute of Lifelong Learning, University of Delhi pg. 5
4.
0 0 0 0
00
( , ) ( , )
lim ( , ) lim ( , ) lim ( , )
x y x y x x y y
f x y f x y f x y
? ? ?
? ? ? ?
4.1. Non-Existence of Limit:
From the simultaneous limit postulates it is amply clear that if
00
( , ) ( , )
lim ( , )
x y x y
f x y
?
? and if () yx ? ? is any function such that
00
( ) when x y x x ???
. Then
? ?
0
lim , ( )
xx
f x x ?
?
, must exist and be equal to . Thus, if we can find two
functions
12
( ), ( ) xx ?? such that the limit of ? ?
1
, ( ) f x x ? and ? ?
2
, ( ) f x x ? are
different, then the simultaneous limit in question does not exist.
Example 1: For the function
? ?
22
,
xy
f x y
xy
?
?
. Find the limit when
( , ) (0, 0) xy ? .
Solution: Let
1
y mx ? , then
1
( , ) (0, 0) 0
lim ( , ) lim ( , )
x y x
f x y f x mx
??
?
2
1
2 2 2
0
1
1
2
0
1
1
2
1
lim
lim
1
1
x
x
mx
x m x
m
m
m
m
?
?
?
?
?
?
?
?
Now, if we take
2
y m x ? , then
2
( , ) (0, 0) 0
lim ( , ) lim ( , )
x y x
f x y f x m x
??
?
2
2
2 2 2
0
2
lim
x
mx
x m x
?
?
?
2
2
0
2
lim
1
x
m
m
?
?
?
2
2
2
1
m
m
?
?
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FAQs on Lecture 8 - Limits and Continuity of Functions of several Variables - Calculus - Engineering Mathematics
| 1. What is the definition of a limit of a function of several variables? |  |
Ans. The limit of a function of several variables is defined as the value that the function approaches as the independent variables approach a certain point in the domain. It can be denoted as:
lim (x, y) → (a, b) f(x, y) = L, where (a, b) is the point in the domain and L is the limit value.
| 2. How do you determine if a limit of a function of several variables exists? | |
Ans. To determine if a limit of a function of several variables exists, we need to check if the function approaches the same value regardless of the direction of approach. This can be done by evaluating the limit along different paths approaching the point. If the limit values along all paths are the same, then the limit exists. However, if the limit values differ along different paths, then the limit does not exist.
| 3. What are the conditions for a function of several variables to be continuous at a point? | |
Ans. A function of several variables is continuous at a point (a, b) if the following conditions are satisfied:
- The function is defined at (a, b).
- The limit of the function as (x, y) approaches (a, b) exists.
- The limit value is equal to the value of the function at (a, b).
If all these conditions are met, then the function is continuous at the given point.
| 4. How can we prove the continuity of a function of several variables? | |
Ans. To prove the continuity of a function of several variables at a point (a, b), we need to show that the function satisfies the conditions for continuity. This can be done by evaluating the limit of the function as (x, y) approaches (a, b) and comparing it with the value of the function at (a, b). If the limit value is equal to the function value, then the function is continuous at the point. Additionally, we can also use the epsilon-delta definition of continuity to provide a formal proof.
| 5. Can a function of several variables be continuous at a point but not differentiable? | |
Ans. Yes, a function of several variables can be continuous at a point but not differentiable. Continuity only requires the function to have a limit at the point and the limit value to be equal to the function value. On the other hand, differentiability requires the existence of partial derivatives of the function at the point. Therefore, it is possible for a function to be continuous but fail to have partial derivatives, making it not differentiable at that point.
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