L' Hopital rule in calculas

1. L’ Hôpital Rule
Presented By: Ali Raza Saleem
Reg. No. : 14-arid-1402

2. Mathematician
• Guillaume François Antoine,
Marquis de L‘ Hôpital
(1661-1704)
• The Rule was Proved by
“John Bernoulli”
• Appear in the Book on
Differential Calculus written
by L‘ Hôpital.

3. Definition
• L‘ Hôpital Rule is a method of
Computing the Limit of indeterminate
forms/functions.
• It is pronounced as Lo-ppi-talls rule.

4. Mathematical Expression
Suppose that and are differentiable
functions on an open interval containing x=a,
(except possibly at x=a), and that is
 lim 0
x a
f x

  lim 0
x a
g x


 lim
x a
f x

   lim
x a
g x

 
 
 
 
 
lim lim
x a x a
f x f x
g x g x 



Above statement is also true for x→a- or x→a+ and
x→ + ∞ or x→ - ∞.
Or
Then
f g

5. Indeterminate Forms
The Equations whose Limits cannot be
determined or the form in which numerator and
denominator both equals to 0 or ∞. Or the
following forms are called indeterminate forms.
Indeterminate forms can be equal to


0
0
0 0
0 1 0


6. Steps to Applying L’Hôpital Rule
STEP 1.
Check that Limit of Given Expression or f (x)/g (x)
is of indeterminate form. If it is not then
L’Hôpital’s Rule cannot be used.
STEP 2.
Differentiate f and g separately.
STEP 3.
Find the limit of . If this limit is finite,
+∞ or -∞ then it is equal to the limit of
   /f x g x 
   /f x g x

7. Limitations
L’Hôpital’s Rule cannot be applied to the finite
limit or to the limit which can be solved simply.
Applying L’Hôpital’s Rule to such limits results in
wrong answers.
For example: Consider
0
6
lim
2x
x
x



8. Limitations cont..
If we solve the above limit then
Answer comes to be 3.
But if we apply L’Hôpital’s Rule to the above limit then
Answer comes to be 1, which is wrong.
0
6 0 6 6
lim 3
2 0 2 2x
x
x
 
  
 
 
 
0 0
6
6
lim lim
2 2
x x
d
x
x dx
dx x
dx
 



  0
1
lim
1x
 1

9. Repetition
• On applying L’Hôpital’s Rule one time, If we
get another indeterminate form then
L’Hôpital’s Rule can be applied more times till
we get an determinate form and then we apply
limit on that expression and we get our
answer.

10. Examples
1. (indeterminate form of type )
Apply L’Hôpital’s Rule..
2
2
4
lim
2x
x
x


4 4 0
2 2 0



 
2
2
4
lim
2
x
d
x
dx
d
x
dx

  
 2
2
lim
1x
x


2
lim 2
x
x


 2 2 4
0
0

11. Examples cont..
 
 0
ln sin
lim
ln tanx
x
x
 
 
ln 0
ln 0





1. Indeterminate form of type
Apply L’Hôpital’s Rule..
 
  20
1/ sin cos
lim
1/ tan secx
x x
x x 20
cos tan
lim
sin secx
x x
x x
  20
cos sin 1
lim
sin cos secx
x x
x x x
  
2
0
limcos
x
x

 2
cos 0 0