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8.7 L’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 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
What makes an expression indeterminate?
lim1000x
x
Consider:
We can hold one part of the expression constant:
1000lim 0x x
There are conflicting trends here. The actual limit will depend on the rates at which the numerator and denominator approach infinity, so we say that an expression in this form is indeterminate.
0
0lim1000 1x
x
Consider:
0.1limxx
0.1lim 0
xx
Finally, here is an expression that looks like it might be indeterminate :
0
lim .1 0x
x
Consider:
We can hold one part of the expression constant:
lim .1 0x
x
The limit is zero any way you look at it, so the expression is not indeterminate.
1000
0lim 0xx
0
1 00 0
0
0
Indeterminate Forms
2
2
4lim
2x
x
x
limx a
f x
g x
f x
g x f x
g x
As 2x
becomes:
The limit is the ratio of the numerator over the denominator as x approaches 2.
2 4x
2x
dx
dxdf
xdgd
df
dg
df
dg
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 or , then:
limx a
f x
g x
lim limx a x a
f x f x
g x g x
0
0
We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:
f a
g a
lim
lim
x a
x a
f x f a
x ag x g a
x a
limx a
f x f a
x ag x g a
x a
limx a
f x f a
g x g a
0lim
0x a
f x
g x
limx a
f x
g x
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!
Example
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.)
1lim sinx
xx
This approaches0
0
1sin
lim1x
x
x
This approaches 0
2
2
1 1cos
lim1x
x x
x
1sin
lim1x
x
x
1lim cosx x
cos 0 1
Example
1
1 1lim
ln 1x x x
1
1 lnlim
1 lnx
x x
x x
Now it is in the form0
0
This is indeterminate form
1
11
lim1
ln ( 1)x
x
x xx
L’Hôpital’s rule applied once.
0
0Fractions cleared. Still
1
1lim
ln 1x
x
x x x
1
1lim
1 1 lnx x
L’Hôpital again.
1
2
Example
Indeterminate Forms: 1 00 0
Evaluating these forms requires a mathematical trick to change the expression into a fraction.
ln lnnu n u
When we take the log of an exponential function, the exponent can be moved out front.
ln1u
n
We can then write the expression as a fraction, which allows us to use L’Hôpital’s rule.
limx a
f x
ln limx a
f xe
lim lnx a
f xe
We can take the log of the function as long as we exponentiate at the same time.
Then move the limit notation outside of the log.
1/lim x
xx
1/lim ln x
xx
e 1
lim lnx
xxe
lnlimx
x
xe 1
lim
1x
x
e 0e 1
0
L’Hôpitalapplied
Example
