Lecture 5 - Indeterminate Forms: L' Hospital Rule | Calculus - Engineering Mathematics PDF Download

 Page 1


Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Indeterminate Forms: L' Hospital Rule 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University 
of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Page 2


Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Indeterminate Forms: L' Hospital Rule 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University 
of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.2 
 
Table of Contents 
 Chapter: Indeterminate Forms: L' Hospital Rule 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: L' Hospital's Rule 
o 3.1: L' Hospital's First Rule (Form 
0
0
) 
o 3.2: L'Hospital Second Rule (Form 
?
?
 ) 
? 4: Indeterminate of the Type 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ?  
o 4.1. Indeterminate Form of the Type 0. ? 
o 4.2. Indeterminate Form of the Type ? ? ? 
o 4.3. Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? 5: Power-Series Method versus L'Hospital Rule 
? Exercises 
? Summary 
? References 
 
 
 
  
Page 3


Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Indeterminate Forms: L' Hospital Rule 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University 
of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.2 
 
Table of Contents 
 Chapter: Indeterminate Forms: L' Hospital Rule 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: L' Hospital's Rule 
o 3.1: L' Hospital's First Rule (Form 
0
0
) 
o 3.2: L'Hospital Second Rule (Form 
?
?
 ) 
? 4: Indeterminate of the Type 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ?  
o 4.1. Indeterminate Form of the Type 0. ? 
o 4.2. Indeterminate Form of the Type ? ? ? 
o 4.3. Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? 5: Power-Series Method versus L'Hospital Rule 
? Exercises 
? Summary 
? References 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.3 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? L' Hospital's First Rule (Form 
0
0
) 
? L'Hospital Second Rule (Form 
?
?
 ) 
? Indeterminate Form of the Type 0. ? 
? Indeterminate Form of the Type ? ? ? 
? Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? Power-Series Method versus L'Hospital Rule 
  
2. Introduction: 
While calculating the limits of the functions at a specified points, we came across the 
situations where we have to calculate the limits of the quotients of the functions such as 
 
0
()
lim
()
xx
fx
gx
?
  
by the algebra of limits, we know that 
 
0
00
0
lim ( )
()
lim , provided lim ( ) 0
( ) lim ( )
xx
x x x x
xx
fx
fx
gx
g x g x
?
??
?
?? 
But what if both f(x) and g(x) tends to 0 (or ?) as 
0
xx ? . 
Thus, in this case limit cannot be calculate and we get a quantity of the form 
 
0
0
or
?
?
  
Thus, the limits of such quotients 
()
()
fx
gx
, where 
 
0
0
0
lim ( )
( ) 0
lim
( ) lim ( ) 0
xx
xx
xx
fx
fx
g x g x
?
?
?
?
? ? ?
?
  
are called indeterminate forms because in this case the limit may not exist or may be any 
real value depending on the particular functions f(x) and g(x). 
There are also occur other limiting situations involving two functions which also generate 
ambiguous values. These indeterminate are indicated by the symbols 
Page 4


Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Indeterminate Forms: L' Hospital Rule 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University 
of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.2 
 
Table of Contents 
 Chapter: Indeterminate Forms: L' Hospital Rule 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: L' Hospital's Rule 
o 3.1: L' Hospital's First Rule (Form 
0
0
) 
o 3.2: L'Hospital Second Rule (Form 
?
?
 ) 
? 4: Indeterminate of the Type 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ?  
o 4.1. Indeterminate Form of the Type 0. ? 
o 4.2. Indeterminate Form of the Type ? ? ? 
o 4.3. Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? 5: Power-Series Method versus L'Hospital Rule 
? Exercises 
? Summary 
? References 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.3 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? L' Hospital's First Rule (Form 
0
0
) 
? L'Hospital Second Rule (Form 
?
?
 ) 
? Indeterminate Form of the Type 0. ? 
? Indeterminate Form of the Type ? ? ? 
? Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? Power-Series Method versus L'Hospital Rule 
  
2. Introduction: 
While calculating the limits of the functions at a specified points, we came across the 
situations where we have to calculate the limits of the quotients of the functions such as 
 
0
()
lim
()
xx
fx
gx
?
  
by the algebra of limits, we know that 
 
0
00
0
lim ( )
()
lim , provided lim ( ) 0
( ) lim ( )
xx
x x x x
xx
fx
fx
gx
g x g x
?
??
?
?? 
But what if both f(x) and g(x) tends to 0 (or ?) as 
0
xx ? . 
Thus, in this case limit cannot be calculate and we get a quantity of the form 
 
0
0
or
?
?
  
Thus, the limits of such quotients 
()
()
fx
gx
, where 
 
0
0
0
lim ( )
( ) 0
lim
( ) lim ( ) 0
xx
xx
xx
fx
fx
g x g x
?
?
?
?
? ? ?
?
  
are called indeterminate forms because in this case the limit may not exist or may be any 
real value depending on the particular functions f(x) and g(x). 
There are also occur other limiting situations involving two functions which also generate 
ambiguous values. These indeterminate are indicated by the symbols 
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.4 
 
 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ? etc. 
These notations correspond to the indicated limiting behavior and juxtaposition of the 
functions f(x) and g(x). These indeterminate are evaluated by using algebraic manipulations, 
logarithms or exponentials to convert them to the basic form 
0
0
or
?
?
. 
I.Q. 1 
3. L' Hospital's Rule: 
The French mathematician Marquis Francois L' Hospital (1661-1704) published the first 
Calculus Book, L' Analyse des infiniment petits, published in 1696. The limit theorem that 
became known as L' Hospital's Rule first appeared in this book, though in fact it was 
discovered by Bernoulli. 
The initial theorem was refined and extended and the various results are collectively 
referred to as L' Hospital's ˆ (or L'Hopital's) rules. 
3.1. L' Hospital's First Rule (Form 
0
0
): 
Theorem 1: Let f(x) and g(x) be defined on [a, b], let ( ) ( ) 0 ( ) 0 f a g a and let g x ? ? ? for 
a < x < b. If f(x) and g(x) are differentiable at a and if '(a) 0 g ? then the limit of 
()
()
fx
gx
 
exists at a and is equal to 
'( )
'( )
fa
ga
. Thus, 
( ) '(a)
lim
( ) '(a)
xa
f x f
g x g
?
?
? . 
Proof: Since 
 ( ) ( ) 0 f a g a ?? 
Thus, we can write the quotient 
()
()
fx
gx
 for a < x < b as follows: 
 
 
 
 
 
Page 5


Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Indeterminate Forms: L' Hospital Rule 
Course Developer: Brijendra Yadav 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University 
of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.2 
 
Table of Contents 
 Chapter: Indeterminate Forms: L' Hospital Rule 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: L' Hospital's Rule 
o 3.1: L' Hospital's First Rule (Form 
0
0
) 
o 3.2: L'Hospital Second Rule (Form 
?
?
 ) 
? 4: Indeterminate of the Type 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ?  
o 4.1. Indeterminate Form of the Type 0. ? 
o 4.2. Indeterminate Form of the Type ? ? ? 
o 4.3. Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? 5: Power-Series Method versus L'Hospital Rule 
? Exercises 
? Summary 
? References 
 
 
 
  
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.3 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? L' Hospital's First Rule (Form 
0
0
) 
? L'Hospital Second Rule (Form 
?
?
 ) 
? Indeterminate Form of the Type 0. ? 
? Indeterminate Form of the Type ? ? ? 
? Indeterminate forms of the Types 
00
0 , 1 ,
?
? 
? Power-Series Method versus L'Hospital Rule 
  
2. Introduction: 
While calculating the limits of the functions at a specified points, we came across the 
situations where we have to calculate the limits of the quotients of the functions such as 
 
0
()
lim
()
xx
fx
gx
?
  
by the algebra of limits, we know that 
 
0
00
0
lim ( )
()
lim , provided lim ( ) 0
( ) lim ( )
xx
x x x x
xx
fx
fx
gx
g x g x
?
??
?
?? 
But what if both f(x) and g(x) tends to 0 (or ?) as 
0
xx ? . 
Thus, in this case limit cannot be calculate and we get a quantity of the form 
 
0
0
or
?
?
  
Thus, the limits of such quotients 
()
()
fx
gx
, where 
 
0
0
0
lim ( )
( ) 0
lim
( ) lim ( ) 0
xx
xx
xx
fx
fx
g x g x
?
?
?
?
? ? ?
?
  
are called indeterminate forms because in this case the limit may not exist or may be any 
real value depending on the particular functions f(x) and g(x). 
There are also occur other limiting situations involving two functions which also generate 
ambiguous values. These indeterminate are indicated by the symbols 
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.4 
 
 
00
0. , 0 , 1 , , ,
??
? ? ? ? ? ? etc. 
These notations correspond to the indicated limiting behavior and juxtaposition of the 
functions f(x) and g(x). These indeterminate are evaluated by using algebraic manipulations, 
logarithms or exponentials to convert them to the basic form 
0
0
or
?
?
. 
I.Q. 1 
3. L' Hospital's Rule: 
The French mathematician Marquis Francois L' Hospital (1661-1704) published the first 
Calculus Book, L' Analyse des infiniment petits, published in 1696. The limit theorem that 
became known as L' Hospital's Rule first appeared in this book, though in fact it was 
discovered by Bernoulli. 
The initial theorem was refined and extended and the various results are collectively 
referred to as L' Hospital's ˆ (or L'Hopital's) rules. 
3.1. L' Hospital's First Rule (Form 
0
0
): 
Theorem 1: Let f(x) and g(x) be defined on [a, b], let ( ) ( ) 0 ( ) 0 f a g a and let g x ? ? ? for 
a < x < b. If f(x) and g(x) are differentiable at a and if '(a) 0 g ? then the limit of 
()
()
fx
gx
 
exists at a and is equal to 
'( )
'( )
fa
ga
. Thus, 
( ) '(a)
lim
( ) '(a)
xa
f x f
g x g
?
?
? . 
Proof: Since 
 ( ) ( ) 0 f a g a ?? 
Thus, we can write the quotient 
()
()
fx
gx
 for a < x < b as follows: 
 
 
 
 
 
Indeterminate Forms: L' Hospital Rule 
 
Institute of Lifelong Learning, University of Delhi                                                 pg.5 
 
 
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
()
( ) ( )
()
f x f x f a
g x g x g a
f x f a
xa
g x g a
xa
?
?
?
?
?
?
?
?
  
Thus applying the limits, we have 
 
( ) ( )
() ()
lim lim
( ) ( )
()
()
( ) ( )
lim
()
( ) ( )
lim
()
'(a)
'(a)
x a x a
xa
xa
f x f a
fx xa
g x g a
gx
xa
f x f a
xa
g x g a
xa
f
g
??
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
 
?  
( ) '(a)
lim
( ) '(a)
xa
f x f
g x g
?
?
? . 
Theorem 2: Let f(x) and g(x) be continuous on [a, b] and differentiable on ]a, b[. Let 
0
[ , ] x a b ? be such that 
00
( ) 0 ( ) f x g x ?? . Let 
0
*( , ) [ , ] S N x a b ? ? ? and R ? ? and 
suppose '( ) 0 . g x x S ? ? ? We have 
 If 
0
'( )
lim
'( )
xx
fx
gx
?
? ? 
 then 
0
()
lim
()
xx
fx
gx
?
? ?. 
Proof: Non-Sequential Approach: 
Given that f(x) and g(x) are continuous and 
00
( ) ( ) 0 f x g x ?? , thus, 
 
00
lim ( ) limg( ) 0
x x x x
f x x
??
?? 
Let 
00
] , [ x x x ? ?? 
Then f(x) and g(x) are smooth on 
0
[ , ] xx . Hence Cauchy mean value theorem applies. As 
such, there exists 
0
] , [ s x x ? such that