Intermediate Forms and L’Hospital Rule (also L’Hopital and Bernoulli Rule)
Guillaume De l'Hôpital 1661 - 1704 8.2 L’Hôpital’s Rule Actually, L’Hôpital’s Rule was...
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Transcript of Guillaume De l'Hôpital 1661 - 1704 8.2 L’Hôpital’s Rule Actually, L’Hôpital’s Rule was...
Guillaume De l'Hôpital1661 - 1704
8.2L’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.
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
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
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 .
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.
Let’s look at another one:
0
0lim1000 1x
x
Consider:
We can hold one part of the expression constant:
0.1limxx
Once again, we have conflicting trends, so this form is indeterminate.
0.1lim 0xx
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
Here is the standard list of the seven indeterminate forms:
0 1 00 0
0
0
There are other indeterminate forms using complex numbers, but those are beyond the scope of this class.
These expressions have to be changed to a fraction before using L’Hopital’s Rule!!!
These expressions can be evaluated “as is” using L’Hopital’s Rule.
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: