L'hospitalsrule

L'Hospital's Rule general presentation.

1. L’HOSPITAL’S RULE
Presented By:
Balaj Khan
Syed kumail

2. Guillaume De l'Hôpital
1661 - 1704
Indeterminate forms
and L’Hospital’s Rule

3. Zero divided by zero can not be evaluated. The limit may
or may not exist, and is called an indeterminate form.
2
2
4
lim
2
x
x
x
®
-
-
Consider: or
If we try to evaluate by direct substitution, we get:
0
0
In the case of the first limit, we can evaluate it by
factoring and canceling:
2
2
4
lim
2
x
x
x
®
-
-
( )( )
2
2 2
lim
2
x
x x
x
®
+ -
=
-
( )
2
lim 2
x
x
®
= + 4=
1
ln
lim
1-®x
x
x
This method does not work in the case of the second limit.
Indeterminate forms

4. Suppose f and g are differentiable and g’(x) ≠ 0 near a (except possible at
a). Suppose that
L’Hospital’s Rule
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limlim
limlim
limlim
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0
0
-¥®¥®®®
®
¥¥
¢
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=
¥¥
±¥=±¥=
==
+-
®®
®®
®®
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xg
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xgxf
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axax
axax

5. L’Hospital’s Rule: Examples
2
2
4
lim
2
x
x
x
®
-
=
-
()
()
lim
x a
f x
g x
®
( )
( )
2
2
4
lim
2
x
d
x
dx
d
x
dx
®
-
=
-
2
2
lim
1
x
x
®
= 4=
1
1
1
1
)1(
)(ln
1
ln
limlimlimlim
1111
===
-
=
- ®®®® x
x
x
dx
d
x
dx
d
x
x
xxxx
¥===
¥®¥®¥® 1
)(
)(
limlimlim
x
x
x
x
x
x
e
x
dx
d
e
dx
d
x
e

6. Example:
2
0
1 cos
lim
x
x
x x
®
-
+
0
sin
lim
1 2
x
x
x
®
=
+
0=
If it’s no longer
indeterminate, then
STOP differentiating!
If we try to continue with L’Hôpital’s rule:
0
sin
lim
1 2
x
x
x
®
=
+
0
cos
lim
2
x
x
®
=
1
2
=
which is wrong!
®

7. On the other hand, you can apply L’Hôpital’s rule as
many times as necessary as long as the fraction is still
indeterminate:
2
0
1 1
2
lim
x
x
x
x
®
+ - -
( )
1
2
0
1 1
1
2 2
lim
2
x
x
x
-
®
+ -
=
0
0
0
0
0
0
not
( )
1
2
2
0
1
1 1
2
lim
x
x x
x
®
+ - -
( )
3
2
0
1
1
4
lim
2
x
x
-
®
- +
=
1
4
2
-
=
1
8
=-

8. 1
lim sin
x
x
x
®¥
æ ö
ç ÷
è ø
This approaches
0
0
1
sin
lim
1x
x
x
®¥
This approaches0¥×
We already know that
0
sin
lim 1
x
x
x
®
æ ö
=
ç ÷
è ø
but if we want to use L’Hôpital’s rule:
2
2
1 1
cos
lim
1x
x x
x
®¥
æ ö æ ö
× -
ç ÷ ç ÷
è ø è ø
=
-
1
sin
lim
1x
x
x
®¥
1
limcos
x x
®¥
æ ö
=
ç ÷
è ø
()cos 0= 1=
Indeterminate Products
Rewrite as a ratio!

9. 1
1 1
lim
ln 1
x x x
®
æ ö
-
ç ÷
-è ø
If we find a common denominator and subtract, we get:
( )
1
1 ln
lim
1 ln
x
x x
x x
®
æ ö- -
ç ÷
ç ÷
-
è ø
Now it is in the form
0
0
¥-¥This is indeterminate form
1
1
1
lim
1
ln
x
x
x
x
x
®
æ ö
-
ç ÷
ç ÷
-
ç ÷+ç ÷
è ø
L’Hôpital’s rule applied once.
0
0
Fractions cleared. Still
1
1
lim
1 ln
x
x
x x x
®
-æ ö
ç ¸
- +è ø
Indeterminate Differences
Rewrite as a ratio!
1
1
lim
1 1 ln
x x
®
æ ö
ç ¸
+ +è ø
L’Hôpital again.
1
2
Answer:

10. Indeterminate Forms:1
¥ 0
0
0
¥
Evaluating these forms requires a mathematical trick to
change the expression into a ratio.
ln ln
n
u n u=
ln
1
u
n
=
We can then write the
expression as a ratio,
which allows us to use
L’Hôpital’s rule.
()lim
x a
f x
®
()( )
ln lim
x a
f x
e
®
=
()( )limln
x a
f x
e
®
=
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 Powers

11. Indeterminate Forms:1
¥ 0
0
0
¥
1/
lim
x
x
x
®¥
( )
1/
limln
x
x
x
e
®¥
()
1
lim ln
x
x
x
e
®¥
()ln
lim
x
x
x
e
®¥
1

lim
1x
x
e
®¥
0
e
1
0
¥
¥
¥
L’Hôpital
applied
Example: