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7/28/2019 15. L'Hpital's Rule
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Guillaume De l'Hpital
1661 - 1704
LHpitals Rule
Actually, LHpitals Rule wasdeveloped by his teacher Johann Bernoulli. De lHpital
paid Bernoulli for privatelessons, and then published thefirst Calculus book based on
those lessons.
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Johann Bernoulli
1667 - 1748
LHpitals Rule
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Zero divided by zero can not be evaluated, and is anexample of an indeterminate form .
2
2
4lim
2 x
x
xConsider:
If we try to evaluate this by direct substitution, we get:00
In this case, we can evaluate this limit by factoring andcanceling:
2
2
4lim
2 x x x
2
2 2lim
2 x x x
x
2
lim 2 x
x 4
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If we zoom in far enough,the curves will appear as
straight lines.
2
2
4lim
2 x
x
xThe limit is the ratio of the numerator
to the denominator
as
x approaches 2.
-5
-4
-3
-2
-10
1
2
3
4
-3 -2 -1 1 2 3x
2 4 x
2 x -0.05
0
0.05
1.95 2 2.05x
lim x a
f x
g x
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2
2
4lim
2 x
x
x
lim x a
f x
g x
-0.05
0
0.05
1.95 2 2.05x
f x
g x
f x
g x
As 2 x
becomes:
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2
2
4lim
2 x
x
x
lim x a
f x
g x
-0.05
0
0.05
1.95 2 2.05x
As 2 x
f x
g x
becomes:
df
dg
df dg
dx
dx
df
xdgd
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2
2 4lim 2 x x x lim x a f x
g x
2
2
4lim
2 x
d x
dxd x
dx
22lim1 x x 4
LHpitals Rule:
If is indeterminate, then:
lim x a
f x
g x
lim lim x a x a
f x f x
g x g x
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We can confirm LHpitals 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
lim x a
f x f a
x ag x g a
x a
lim x a
f x f a
g x g a
0lim
0 x a f x
g x
lim x a
f x
g x
Let f and g be defined on an interval where f(x) 0 and
g(x) 0 as x a, but tends to a finite limit L.
xg
x f
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Example:
20
1 coslim x
x x x 0
sinlim
1 2 x x x
0
If its no longer indeterminate, thenSTOP !
If we try to continue with LHpitals rule:
0
sinlim
1 2 x
x
x0
coslim
2 x
x 1
2
which is wrong,
wrong, wrong!
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On the other hand, you can apply LHpitals rule asmany times as necessary as long as the fraction is still
indeterminate:
20
1 12lim
x
x x
x
12
0
1 11
2 2lim 2 x
x
x
0
0
0
0
0
0not
12
20
11 1
2lim x
x x
x
32
0
11
4lim 2 x
x 14
2
1
8
(Rewritten inexponentialform.)
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When applying LHpitals
Rule, one has to differentiate the
numerator and the denominator separately .
A common mistake is to differentiate the whole expression f(x
) / g(x)
This leads to tedious, unnecessary, and, most importantly,
wrong computations.
Correct calculation:
2
1
2cos2lim
2sinlim
2sinlim
000
x
xdxd
xdxd
x
x x x x
Incorrect calculation:
????2sin2cos2lim2sinlim 200 x x x x
x x
x x
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LHpitals rule can be used to evaluate other indeterminate
00
forms besides .
The following are also considered indeterminate:
0 10
00
The first one, , can be evaluated just like .0
0The others must be changed to fractions first.
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What makes an expression indeterminate?
lim 1000 x x
Consider:
We can hold one part of the expression constant:
1000lim 0 x x
There are conflicting trends here. The actual limitwill depend on the rates at which the numerator anddenominator approach infinity, so we say that anexpression in this form is indeterminate .
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Lets look at another one:
0
0lim1000 1 x
x
Consider:
We can hold one part of the expression constant:
0.1
lim x x
Once again, we have conflicting trends, so this formis indeterminate .
0.1
lim 0 x x
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Here is an expression that looks like it might be
indeterminate :
0
lim .1 0 x
x
Consider:
We can hold one part of the expression constant:
lim .1 0 x
x
The limit is zero any way you look at it, so theexpression is not
indeterminate .
1000
0lim 0 x x
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Finally, here is an expression that looks like it
should NOT be indeterminate :
We can hold one part of the expression constant:
Consider: 1
461000411000
100010001000
-101.759.0;102.471.1
:
01
lim;11;1
lim
ex
x x
x x
Once again, we have conflicting trends, so this form is
indeterminate .
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Here is the standard list of indeterminate forms:
0
1 00 0
00
There are other indeterminate forms using complexnumbers, but those are beyond the scope of this class.
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Example
x 1
1 1
x 1 1
Let's return to our former example: lim1
1. Check to see if L'Hopital's rule applies:lim( ) 0 and lim( 1) 0
L'Hopital's rule does apply.
( )lim lim lim
1 ( 1)
x
x
x x
x x
x
e e x
Stepe e x
d e ee e dx
d x xdx
1
1
x
x
ee
a
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Example:
1lim
1lim
1lim 2
2
2
2
2 x x
x
x
x
x x x x
1lim
2 x
x x
;1
lim1
lim 22
2
2
x x
x x
x x
Determine the limit:
Rewrite:
Use the Power Law: The limit of a functions positive integer power is the power of the functions limit:
OR Ratio of the leading powers:
1
1lim
1lim
2
2
2
2
xdxd
xdxd
x x
x x
111
1lim
1lim 2
2
2
2
x x
x x
x x
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1lim sin x
x x
This approaches00
1sin
lim
1 x
x
x
This approaches 0
We already know that0
sinlim 1 x
x
x
but if we want to use LHpitals rule:
2
2
1 1cos
lim1 x
x x
x
1sinlim
1 x x
x
1lim cos x x
cos 0 1
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1
1 1lim
ln 1 x x x
If we find a common denominator and subtract, we get:
11 lnlim1 ln x
x x x x
Now it is in the form00
This is indeterminate form
1
11lim
1ln
x
x x
x
x
LHpitals rule applied once.
0
0
Fractions cleared. Still1
1lim
1 ln x x
x x x
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1
1 1lim
ln 1 x x x
11 lnlim1 ln x
x x x x
1
11lim
1ln
x
x x
x x
1
1lim 1 1 ln x x
LHpital
again.
1
2
1
1lim
1 ln x
x
x x x
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Cautionary Example
1
1 1
1
lnExample: Evaluate lim
Step 1. Check to see if L'Hopital/s rule applies:
lim ln ln1 0 but lim 1 0
L'Hopital's Rule does not apply.
Use the quotient property for limits instead:limln
lim
x
x x
x
x
x x
x x
x x
1
1
ln 00
lim 1
Using L'Hopital's Rule would have given us
an incorrect result.
x
x
x
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Indeterminate Forms: 1 00 0
Evaluating these forms requires a mathematical trick tochange 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 theexpression as a fraction,which allows us to useLHpitals
rule.
lim x a
f x ln lim
x a f x
e
lim ln x a
f x
e
We can take the log of the function as long
as we exponentiate
at the same time.
Then move thelimit notationoutside of the log.
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Indeterminate Forms: 1 00 0
1/lim x x
x
1/lim ln x x xe
1lim ln x
x xe
lnlim
x
x
x
e1
lim 1 x x
e0
e1
0
LHpitalapplied
Example: