Post on 01-Jan-2016
ExtraL’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
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
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 indeterminate
0
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.
1
2x lim
2
1
x
xx
xx
11x lim
0
x
5xsin lim
0
x
3
64x lim
2
1
x
xx
xx
4)-ln(x lim
x
xx
73x lim
2
0
2
)4sin( lim
2
2
x
xx