ExtraL’Hôpital’s Rule

Zero divided by zero can not be evaluated, and is an example of indeterminate form.

2

2

4lim

2x

x

x

Consider:

If we try to evaluate this by direct substitution, we get:0

0

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

2

2

4lim

2x

x

x

2

2 2lim

2x

x x

x

2lim 2x

x

4

If we zoom in far enough, the curves will appear as straight lines.

2

2

4lim

2x

x

x

The limit is the ratio of the numerator over the denominator as x approaches 2.

2 4x

2x

limx a

f x

g x

2

2

4lim

2x

x

x

limx a

f x

g x

2

2

4lim

2x

dx

dxdx

dx

2

2lim

1x

x

4

L’Hôpital’s Rule:

If is indeterminate, then:

limx a

f x

g x

lim limx a x a

f x f x

g x g x

Example:

20

1 coslimx

x

x x

0

sinlim

1 2x

x

x

0

If it’s no longer indeterminate, then STOP!

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

0

sinlim

1 2x

x

x

0

coslim

2x

x

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:

20

1 12lim

x

xx

x

1

2

0

1 11

2 2lim2x

x

x

0

0

0

0

0

0not

1

2

20

11 1

2limx

x x

x

3

2

0

11

4lim2x

x

14

2

1

8

(Rewritten in exponential form.)

L’Hôpital’s rule can be used to evaluate other indeterminate

0

0forms besides .

The following are also considered indeterminate:

0 1 00 0

The first one, , can be evaluated just like .

0

0

The others must be changed to fractions first.

1

2x lim

2

1

x

xx

xx

11x lim

0

x

5xsin lim

0

x

3

64x lim

2

1

x

xx

xx

4)-ln(x lim

x

xx

73x lim

2

0

2

)4sin( lim

2

2

x

xx