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Title: L'Hopital's Rule
Description: Evaluation of Indeterminate forms of limits. Clear and concise L'Hopital's Rule discussion. Many examples! Mathematical Analysis for College Calculus. Evaluating Limits at Infinity over Infinity or Zero over Zero.
Description: Evaluation of Indeterminate forms of limits. Clear and concise L'Hopital's Rule discussion. Many examples! Mathematical Analysis for College Calculus. Evaluating Limits at Infinity over Infinity or Zero over Zero.
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Unit 1
...
0
Indeterminate Form ∞/∞
f x
A limit of the form lim
where
x a g x
lim f x and lim g x is called
x a
x a
an indeterminate form of type
...
7
...
ln x
1
...
lim
x 1
x
0
0
e x
3
...
L’Hôpital’s Rule
Suppose f and g are differentiable and g ' x 0
near a except possibly at a
...
Then
if the limit on the right side exists or is
...
7
...
ln x
1
...
lim
x 0 1 cos x
L
2x
lim
x 0 sin x
L
2
lim
x 0 cos x
2
0
0
0
0
Arctan x 0
3
...
If lim f x and lim g x 0 then
x a
x a
the limit is called an indeterminate form of
type 0
...
Write fg as a quotient:
f
fg
or
1
g
ˆ
2
...
g
fg
1
f
Example 1
...
3
Evaluate lim x ln x
...
The following are also considered as
...
7
...
x 1 x 1
x 1
2
1
lim
2
x 1 x 1
x 1
x 1
0
lim 2
0
x 1 x 1
L
1
lim
x 1 2 x
1
2
Indeterminate Form of
Type 00, ∞0, and 1∞
For lim f x
x a
g x
,
1
...
lim f x and lim g x 0
type
3
...
7
...
x 0
lim x
x 0
Let
x2
00
yx
x2
ln y ln x
x2
ln y x ln x
2
lim ln y lim x2 ln x
x 0
x 0
ln x
lim
x 0 1
x2
0
lim ln y lim x2 ln x
x 0
x 0
0
ln x
lim
x 0 1
2
x
1
L
x
lim
x 0 2
x3
2
x
lim
x 0
2
0
yx
lim ln y 0
x2
x 0
lim x lim y
x2
x 0
x 0
lim e
x 0
e
1
0
ln y
ye
ln y
Evaluating Indeterminate
Forms of Type 00, ∞0, and 1∞
Given lim f x
g x
x a
1
...
Then ln y g x ln f x
...
Evaluate limln y
...
Use the identity y e
That is, lim f x
x a
ln y
g x
to find lim f x
x a
lim y lim e
...
Example 1
...
6
x
2
Evaluate lim 1
...
x
2
ln y x ln 1
x
2
lim ln y lim x ln 1
x
x
x
2
ln 1
x
lim
x
1
x
1 2
2
2 x
1
L
x
lim
x
1
x2
0
0
0
2
2
2 x
1
L
x
lim
x
1
2
x
2x
lim
x x 2
L
2
lim
x 1
2
1
2
y 1
x
x
lim ln y 2
x
x
2
lim 1 lim y
x
x x
lim eln y
x
e
2
Required Exercises
Answer Exercises in TC7
...
7 (1-28) on page 642-643
Exercises 7
...
End of Unit 1
Title: L'Hopital's Rule
Description: Evaluation of Indeterminate forms of limits. Clear and concise L'Hopital's Rule discussion. Many examples! Mathematical Analysis for College Calculus. Evaluating Limits at Infinity over Infinity or Zero over Zero.
Description: Evaluation of Indeterminate forms of limits. Clear and concise L'Hopital's Rule discussion. Many examples! Mathematical Analysis for College Calculus. Evaluating Limits at Infinity over Infinity or Zero over Zero.