MAT 1234
Calculus I
Section 1.6 Part II
Using the Limit Laws
http://myhome.spu.edu/lauw
Homework
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WebAssign 1.6 Part II
Turn in the one problem at the end of
your handout (the last page) tomorrow.
Recall: Limit of a Function
lim f ( x)  L
xa
if and only if
lim f ( x)  L
xa

Independent of f(a)
and
lim f ( x)  L
xa
Preview
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Limits with Absolute Values
The Squeeze Theorem
Pay extra attention to the presentation
expectations (especially those who have
photographic memory!)
Recall (absolute value function)
 x if x  0
x 
 x if x  0
Example 8(a)
(Limits with Absolute Values)
lim x
x0



This is not a polynomial or an rational
function
Absolute function is not in one of the 11
limit laws
Find the limit by using the definition of
limits
Example 8(a)
(Limits with Absolute Values)
lim x
x0



This is not a polynomial or an rational
function
Absolute function is not in one of the 11
limit laws
Find the limit by using the definition of
limits
Example 8(a)
(Limits with Absolute Values)
lim x
x0



This is not a polynomial or an rational
function
Absolute function is not in one of the 11
limit laws
Find the limit by using the definition of
limits
Example 8(a)
(Limits with Absolute Values)
lim x
x0



This is not a polynomial or an rational
function
Absolute function is not in one of the 11
limit laws
Find the limit by using the definition of
limits
Example 8(a)
(Limits with Absolute Values)
lim x
x0
We are going to look at left and right hand limits:
1.
As x  0 , x  0
lim x 
x 0
2.
As x  0 , x  0
lim x 
x 0
 lim x
x 0
 x if x  0
x 
 x if x  0
Example 8(a)
(Limits with Absolute Values)
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Note that solutions steps are very
important here. You need to pay attention
and understand the correct way of
presenting your solutions.
WebAssign do not check your solutions
steps.
Advise: Write out full solutions in your
notebook even WebAssign does not ask for
it.
Example 8(b)
x2
lim
x 2 x  2
Example 8(b)
x2
lim
x 2 x  2

Again, no direct substitution!!
 x if x  0
x 
 x if x  0
if x  2  0
 x2
x2  
( x  2) if x  2  0
Example 8(b)
1.
As x  2 , x  2
lim
x 2
2.
x2

x2
As x  2 , x  2
lim
x 2
x2

x2
x2
 lim
x 2 x  2
if x  2
 x2
x2  
( x  2) if x  2
Example 9
(Limits with Absolute Values)

1 1
lim  5   

x 0 
x
x


Example 9
(Limits with Absolute Values)

1 1
lim  5   

x 0 
x
x


As x  0 , x  0

1 1
lim  5    
x 0 
x x 

 x if x  0
x 
 x if x  0
Squeeze Theorem
If f ( x)  g ( x)  h( x) when x is near a
and lim f ( x)  lim h( x)  L
x a
then
x a
lim g ( x)  L
x a
Squeeze Theorem
y
f ( x)  g ( x)  h( x)
h(x)
g (x)
f (x)
a
x
Squeeze Theorem
y
f ( x)  g ( x)  h( x)
h(x)
lim f ( x)  lim h( x)  L
xa
L
f (x)
a
x
xa
Squeeze Theorem
y
f ( x)  g ( x)  h( x)
h(x)
lim f ( x)  lim h( x)  L
xa
g (x)
L
xa
lim g ( x )  L
xa
f (x)
a
x
Squeeze Theorem
y
h(x)
g (x)
L
f (x)
a
x
You will see
this type of
idea over and
over again.
Example 10

1
2
lim x sin   lim x
x0
 x  x0
2



 1 
  lim sin  
x 0
 x 

We cannot apply the limit laws since
1
lim sin 
x 0
 x
DNE (Lab 2, 4(c))
Example 10

1
2
lim x sin   lim x
x0
 x  x0
2



 1 
  lim sin  
x 0
 x 

We cannot apply the limit laws since
1
lim sin 
x 0
 x
DNE (Lab 2, 4(c))
Example 10

1
2
lim x sin   lim x
x0
 x  x0
2



 1 
  lim sin  
x 0
 x 

We cannot apply the limit laws since
1
lim sin 
x 0
 x
DNE (Lab 2)
Example 10
1
y  sin  
 x
1
lim x sin  
x 0
x
2
Example 10
1
 sin   
x
f ( x)  g ( x)  h( x)
lim f ( x)  lim h( x)  L
xa
1
 x sin   
x
2
xa
lim g ( x )  L
xa
Expectations
1
 sin   
x
1
 x 2 sin   
x
lim

 x0
and 
lim

 x0
Example 10
1
y  x sin  
x
2
Review: We learned…
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How to find limits involving absolute values.
The idea of the squeeze theorem.
Pay attention to the expectations of the solutions
details.
Note that in the exam, you will be ask to state
the theorems you learned in this class.
Quiz
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Please use pencils and erasers.
Please make sure your handwriting is
easy to understand.
You have 20 min.
Do not look at your neighbor’s paper. If
you need to relax your neck, …
Double check your work.
Once you turn in your paper, you can go.
Quiz
Evaluate lim
h 0
lim
h0
2h  2
h
2h  2

h