Objectives for Section 12.3
L’Hôpital’s Rule
 The student will be able to apply
L’Hôpital’s Rule to the
Indeterminate Form 0/0.

The student will be able to evaluate one-sided limits and limits at

.

The student will be able to apply
L’Hôpital’s Rule to the
Indeterminate Form

/

.
Barnett/Ziegler/Byleen Business Calculus 11e 1

Limits involving Powers of x
In this section we will develop a powerful technique for evaluating limits of quotients called
L’Hôpital’s Rule .
To use this rule, it is necessary to be familiar with the limit properties of some basic functions which follow.
Barnett/Ziegler/Byleen Business Calculus 11e y
 x x x x lim

0 x

0 lim
 x
  lim
 x
  y
 x
2 x x x lim
2
0

0 x
 lim
 x
2   lim
 x
2
 
2

Limits Involving Powers of
(continued) x y

1 x x lim

0

1 x
  x lim

0

1 x
  x

1 lim Does not exist
0 x x
1 lim 0
 x
 x
1 lim 0
 x

Barnett/Ziegler/Byleen Business Calculus 11e y

1 x
2 x lim

0

1 x
2 x lim

0

1 x
2
 
  x

0 x
1 lim
2
 
1 x lim
 x 2
1 x lim
 x
2


0
0
3

Limits Involving Exponential and Logarithmic Functions y
 e x x x x lim

0 e x 
1 lim
 e x   lim
 e x 
0 y
 e
 x lim x

0 e
 x x lim
 e
 x x lim
 e
 x

1

0
 
Barnett/Ziegler/Byleen Business Calculus 11e y
 ln x x x lim ln

0
 x lim ln
 x
 
 
4

L’Hôpital’s Rule and the
Indeterminate Form 0/0 lim x
 c
is a 0/0 indeterminate form if lim ( )
 g x

0.
x
 c x
 c
The quotient property for limits does not apply since lim ( )

0.
x
 c lim x

2 x
2 
4 x

2
is a 0/0 indeterminate form but it can be
ev aluated using algebraic simplification.
The limit lim x

1 e x
 e x

1
cannot be evaluated this way.
Barnett/Ziegler/Byleen Business Calculus 11e 5

L’Hôpital’s Rule and 0/0
(continued)
Limits such as the one on the previous slide can be evaluated using
L’Hôpital’s Rule
:
For c a real number,
If lim ( )
 g x

0 then x
 c x
 c lim x
 c
 lim x
 c provided the second limit exists or
Barnett/Ziegler/Byleen Business Calculus 11e 6

Example
Let's return to our former example: lim x

1 e x  e x

1
Step 1. Check to see if L'Hopital's rule applies: lim( x

1 e x  
0 and lim( x

1 x
 
0
L'Hopital's rule does apply.
lim x

1 e x  e x

1
 lim x

1 d
( e x  e ) dx d dx
( x

1)
 lim x

1 e x
1
 e
Barnett/Ziegler/Byleen Business Calculus 11e 7

Cautionary Example
Example: Evaluate lim x

1 ln x x
Step 1. Check to see if L'Hopital/s rule applies: lim ln x

1 x
 ln1

0 but lim x

1 x
 
0
L'Hopital's Rule does not apply.
Use the quotient property for limits instead: lim x

1 ln x x
 lim x

1 lim x

1 ln x x
1
0
Using L'Hopital's Rule would have given us an incorrect result.
Barnett/Ziegler/Byleen Business Calculus 11e 8

One-Sided Limits and Limits at 
Theorem 2.
(L’Hôpital’s Rule, Version 2 )
The first version of L’Hôpital’s Rule remains valid if the symbol x
 c is replaced everywhere it occurs with one of the following symbols: x
 c + x
 c x
  x

-

Barnett/Ziegler/Byleen Business Calculus 11e 9

x

1

x x
 x  x

1
 x x
2
1
2
Example
Evaluate: lim x

1
+ ln x
( x

1)
2
Step 1. Check to see if L'Hopital's rule applies: lim ln x

0 and lim ( x

1)
2 
0 x

1
+ x

1
+
L'Hopital's rule does apply.
Step 2. Apply L'Hopital's rule: lim x

1
+ ln x
( x

1)
2
 lim x

1
+ d ln x d dx
( x

1)
2 dx
1
 lim x

1
+ x
2 (

1)
 lim x

1
+ 2 (
1

1)
 
The limit as x
 
1 is

because 1/2x(x-1) has a vertical asymptote at x = 1.
Barnett/Ziegler/Byleen Business Calculus 11e 10

L’Hôpital’s Rule and the
Indeterminate Form  / 
Theorem 3.
(L’Hôpital’s Rule, version 3)
Versions 1 and 2 of L’Hôpital’s Rule are also valid if x lim
 c f ( x )
 lim x
 c g ( x )
 
Barnett/Ziegler/Byleen Business Calculus 11e 11

Example
Evaluate lim x
 ln x x
2
Step 1. Check to see if L'Hopital's rule applies: x lim
 ln x x
2
 
and lim x
 x
2  
L'Hopital's rule does apply.
Step 2. Apply L'Hopital's Rule x lim
 ln x x
2
 x lim
 d dx d ln x x
2 dx
1
 x lim
 x
2 x
 x l im
 l
2 x
2

0
Barnett/Ziegler/Byleen Business Calculus 11e 12