if we zoom in far enough, the curves will appear as straight lines. the limit is the ratio of the...
TRANSCRIPT
If we zoom in far enough, the curves will appear as straight lines.
The limit is the ratio of the numerator over the denominator as x approaches a.
limx a
f x
g x
limx a
f x
g x
As x a
f x
g xbecomes:
( )( ) (
( )( ) ( )
)
g a x
f a x a f
g
a
a a
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
Provided that the limit on the right exists
We can confirm L’Hôpital’s rule 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
sinlim1 2x
x
x
0
If it’s no longer indeterminate, then STOP!
If we try to continue with L’Hôpital’s rule:
0
sinlim1 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
142
1
8
(Rewritten in exponential form.)
Guillaume Francois Antoine,Marquis de l'Hôpital
1661 - 1704
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.
l'Hôpital is commonly spelled as both "l'Hospital" and "l'Hôpital." The Marquis spelled his name with an 's'; however, the French language has since dropped the 's' (it was silent anyway) and added a circumflex to the preceding vowel.
Johann Bernoulli1667 - 1748
In 1694 he forged a deal with Johann Bernoulli. The deal was that l'Hôpital paid Bernoulli 300 Francs a year to tell him of his discoveries, which l'Hôpital described in his book. In 1704, after l'Hôpital's death, Bernoulli revealed the deal to the world, claiming that many of the results in l'Hôpital's book were due to him. In 1922 texts were found that give support for Bernoulli. The widespread story that l'Hôpital tried to get credit for inventing de l'Hôpital's rule is false: he published his book anonymously, acknowledged Bernoulli's help in the introduction, and never claimed to be responsible for the rule
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.
L’Hôpital’s rule can be used to evaluate other indeterminate0
0forms besides .
The following are also considered indeterminate:
0
The first one, , can be evaluated just like .
0
0
The others must be changed to fractions first.
3
2lim
x
x
e
x
33lim
2
x
x
e
x Still in the form
This is indeterminate form
39lim
2
x
x
e
Now it is determined.
1lim sinx
xx
This approaches0
0
1sin
lim1x
x
x
This approaches 0
We already know that0
sinlim 1x
x
x
we can confirm this using L’Hôpital’s rule:
2
2
1 1cos
lim1x
x x
x
1sin
lim1x
x
x
1limcosx 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
lim1ln
x
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
lim1ln
x
xx
xx
1
1lim
1 1 lnx x
L’Hôpital again.
1
2
1
1lim
1 lnx
x
x x x
Let’s look at another indeterminate form:
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
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
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.
Here is the standard list of indeterminate forms:
0
1 00 0
0
0
Write as a RATIO
Use LOGS