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Guillaume De l'Hôpital1661 - 1704
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.
Greg Kelly, Hanford High School, Richland, Washington
Johann Bernoulli1667 - 1748
8.1: 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 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, what will the curves begin to look like?
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
Straight lines.
2
2
4lim
2x
x
x
limx a
f x
g x
f x
g x
f x
g x
At x = 2
approaches
approaches its tangent line
0)2)(2( xf
0)2)(2( xg=
f (2)
g (2)
)2(f )2(g=
So how can we use this to help us find the limit here?
its tangent line
2
2
4lim
2x
x
x
limx a
f x
g x
2
2
4lim
2x
dx
dxd
xdx
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
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
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 indeterminate0
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.
1lim sinx
xx
This approaches0
0
1sin
lim1x
x
x
This approaches 0
We already know that0
sinlim 1x
x
x
but if we want to use L’Hôpital’s rule:
2
2
1 1cos
lim1x
x x
x
1sin
lim1x
x
x
1lim cosx x
cos 0 1
1
1 1lim
ln 1x x x
If we find a common denominator and subtract, we get:
1
1 lnlim
1 lnx
x x
x x
Now it is in the form0
0
This is indeterminate form
1
11
lim1
lnx
xx
xx
L’Hôpital’s rule applied once.
0
0Fractions cleared. Still
1
1lim
1 lnx
x
x x x
1
1 1lim
ln 1x x x
1
1 lnlim
1 lnx
x x
x x
1
11
lim1
lnx
xx
xx
1
1lim
1 1 lnx x
L’Hôpital again.
1
2
1
1lim
1 lnx
x
x x x
Indeterminate Forms: 1 00 0
Evaluating these forms requires a mathematical trick to change the expression into a fraction.
lnlim lim lim f x L
x a x a x af x L f x e e
1/lim x
xx
1/lim ln x
xx
e
1lim lnx
xxe
lnlimx
x
xe
1
lim1x
x
e
0e 1
0
L’Hôpitalapplied
