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Homework…
WebAssign 6.8• Part 1 (Friday, 20 problems, 57 min.)• Part 2 (Monday, 6 problems, 22 min.)
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You have some experience with this topic from Lab 01
Definition of Indeterminate Forms When and How to apply the L’Hospital’s
Rule to find limits
A Common Theme in calculus -Limits
Some limits cannot be evaluated by simplifications and substitutions.
1sin
lim0
x
xx
A Common Theme in calculus -Limits
Some limits cannot be evaluated by simplifications and substitutions.
In section 2.4, we use geometric arguments to prove that
1sin
lim0
x
xx
A Common Theme in calculus -Limits
A lot of applications We have this limit in Calculus 1-3. Used to prove the derivative of
1sin
lim0
x
xx
Simple Pendulum
When the angle is small, the motion can be modeled by
l
02
2
l
g
dt
d
2
2sin 0
d g
dt l
3 5 7
sin3! 5! 7!
Definition( )
lim is called an if ( )
either (a) lim ( ) 0 and lim ( ) 0
or (b) lim ( ) and lim ( )
indeterminate for
m
x a
x a x a
x a x a
f x
g x
f x g x
f x g x
Definition0
0 Type
Type
( )lim is called an indeterminate form if
( )
either (a) lim ( ) 0 and lim ( ) 0
or (b) lim ( ) and lim ( )
x a
x a x a
x a x a
f x
g x
f x g x
f x g x
L’Hospital’s Rule
). is(or exists side handright on thelimit theif
, )(
)(lim=
)(
)(lim
Then, form. ateindeterminan is )(
)(lim that Suppose
).at possibly (except near 0)(
and abledifferenti are and Suppose
xg
xf
xg
xf
xg
xf
aaag
gf
axax
ax
Example 1
Step 1: Check that is an indeterminate form x
xx
sinlim
0
sin
0,
x
x
x
Supporting steps should be done on the right hand column.
Remark on non-standard notation
0 0
sin coslim lim
1
H
x x
x x
x
The following non-standard notation is not acceptable in this class.
Remark on non-standard notation
The following non-standard notation is not acceptable in this class.
0 0
sin coslim lim
1
H
x x
x x
x
Remarks
For some problems, you may need to apply the rule more than once. But make sure you check the condition in step 1 every time you apply the rule.
Q&A
Q: Can I apply the l’Hospital’s rule if the limit is not in quotient form?
A: Sometimes, we may rewrite the limit into quotient form and then apply the l’Hospital’s rule.
Geometric Meaning
0 0
sinlim lim cosx x
xx
x
2
1 1
1lim lim 1
1x x
xx
x
sin xy
x
cosy x
2 1
1
xy
x
1y x
Highly Experimental...Battle “Group-lactica”
Do you have what it takes to beat the other study groups?
Criteria: On average, your study group has the highest increase in score in the second exam.
Award: bonus points equal to the 1.0 x average increase in your group to all members in the group. (Higher multiplier possible for dramatic increase.)
Talk to me if you are interested. Must have at least 3 groups to participate.Group size = 3 – 4 students.