8.1: L’Hôpital’s Rule

Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.

Guillaume De l'Hôpital 1661 - 1704

Greg Kelly, Hanford High School, Richland, Washington

8.1: L’Hôpital’s Rule

Johann Bernoulli 1667 - 1748



Consider:

x

lim  2

x

2  4

x

 2 If we try to evaluate this by direct substitution, we get: 0 0 Zero divided by zero can not be evaluated, and is an example of

indeterminate form

.

In this case, we can evaluate this limit by factoring and canceling:

x

lim  2

x

2  4

x

 2 

x

lim  2 

x



x

2 

x

 2  2   lim

x

 2 

x

 2   4 

x

lim 

a

    

x

lim  2

x

2  4

x

 2 The limit is the ratio of the numerator over the denominator as

x

approaches 2.

x

2  4

x

 2 -3 -2 1 -1 0 -1 -2 -3 -4 -5 4 3 2 1 x 2 3 0.05

If we zoom in far enough, the curves will appear as 0 1.95

2.05

straight lines.

-0.05



0.05

x

lim 

a

    

x

lim  2

x

2  4

x

 2 0 1.95

2.05

As    

x

 2 becomes: -0.05



0.05

x

lim 

a

    

x

lim  2

x

2  4

x

 2

df

0 1.95

-0.05

dx dg

2.05

As    

x

 2 becomes:

df dg df



dx dg d x



x

lim 

a

    

x

lim  2

x

2  4

x

 2 

x

lim  2

d



x

2  4 

dx d dx



x

 2  

x

lim  2 2

x

1  4

L’Hôpital’s Rule:

x

lim 

a

   

x

lim 

a

    

x

lim 

a f

     

We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:

f

     

x

lim 

a x

lim 

a

  

x

 

x



a a



x

lim 

a

  

x

 

x



a a



x

lim 

a

          

x

lim 

a

      0 0 

x

lim 

a

    

Example:

x

lim  0

x



x

2

x



x

 sin 

x x

 0 If it’s no longer indeterminate, then

STOP

!

If we try to continue with L’Hôpital’s rule: 

x

 sin 

x x



x

lim  0 cos

x

2  1 2 which is wrong, wrong, wrong!



On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate:

x

lim  0 1

x x

2

x

2 0 0   1 4 2

x

lim  0  1 

x

 1 2

x

2 1 2

x



x

lim  0 1 2  1 

x

  1 2 2

x

 1 2 (Rewritten in exponential form.) 0 0   1 8 

x

lim  0  1 4  1  2

x

  3 2 not 0 0 

L’Hôpital’s rule can be used to evaluate other indeterminate 0 forms besides .

0 The following are also considered indeterminate:    0

  

1  0 0   0 0 The others must be changed to fractions first.

 0 

x

lim 

x

sin 1

x

This approaches  0

x

lim 

x

lim  sin 1

x

1

x

This approaches We already know that

x

lim  0 sin

x x

   1 but if we want to use L’Hôpital’s rule: sin 1

x

1

x



x

lim  cos 1     1

x

2 1

x

2 

x

lim cos  1   0 0   1 

lim

x

 1 1 ln

x



x

1  1 This is indeterminate form

  

If we find a common denominator and subtract, we get: lim

x

 1   

x x



x x

  lim

x

 1      

x

1  1

x

 1  ln

x x

      lim

x

 1 ln

x x

 1 1 Now it is in the form 0 0 L’Hôpital’s rule applied once.

Fractions cleared. Still 0 0 

lim

x

 1 1 ln

x



x

1  1 lim

x

 1   

x x



x x

  lim

x

 1      

x

1  1

x

 1  ln

x x

      lim

x

 1 ln

x x

 1 1 lim

x

 1     1

x

1  l     L’Hôpital again.

1 2 

Indeterminate Forms: 1  0 0  0 Evaluating these forms requires a mathematical trick to change the expression into a fraction.

x

lim 

a x

lim 

x

1/

x e

lim ln

x

 

L

 lim

x



a

 0

e

lim

x

 1

x

ln

e

lim

x

 ln

x



x

lim 

a e

ln  

e

lim

x

 1

x

1

e

0 

e L

L’Hôpital applied  1 p