guillaume de l'hôpital 1661 - 1704 8.2 day 1 l’hôpital’s rule actually, l’hôpital’s...
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Guillaume De l'Hôpital1661 - 1704
8.2 day 1L’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
8.2 day 1L’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
f x
g x f x
g x
As 2x
becomes:
2
2
4lim
2x
x
x
limx a
f x
g x
As 2x
f x
g xbecomes:
df
dg
df
dg
dx
dxdf
xdgd
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
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
sin lim
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.
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.
Indeterminate Forms: 1 00 0
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
Example:
A special symbol for L’Hôpital’s rule :
20
1 coslimx
x
x x
H
0
sin lim
1 2x
x
x 0
Note the letter “H” over the equal sign.
Some textbooks (and some college professors) use aspecial form of the equal sign when L’Hôpital’s rule is applied:
Our textbook does not use this symbol, but be aware that you may see it in the future. You may also see the letters L’H written over the equal sign, which means the same thing.