MAT 3749
Introduction to Analysis
Section 2.1 Part 3
Squeeze Theorem and
Infinite Limits
http://myhome.spu.edu/lauw
Notes
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Group reassignments
Math Party
Exam 1
Please study for the quizzes
Major Themes
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Introduction to proofs in the context of
calculus 1
Make sure future teachers to have a
better understanding of calculus 1
Look at (rigorous) ideas in analysis
which can be extended to more
advanced math
References
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Section 2.1
Preview
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Squeeze Theorem
One-sided Limits
Limits at Infinities
Infinite Limits
Squeeze Theorem
If f ( x)  g ( x)  h( x) in some deleted neighborhood of 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 1

1
2
lim x sin    lim x
x 0
 x  x0
2


 1 
  limsin   
x 0
 x 

Example 1

1
2
lim x sin    lim x
x 0
 x  x0
2


 1 
  limsin   
x 0
 x 

Example 1

1
2
lim x sin    lim x
x 0
 x  x0
2



 1 
  limsin   
x 0
 x 

We cannot apply the limit laws since
1
lim sin 
x 0
 x
DNE (2.1.1)
Example 1
f ( x)  g ( x)  h( x)
1
 sin   
x
lim f ( x)  lim h( x)  L
xa
xa
lim g ( x )  L
xa
Make sure to quote the name of the Squeeze
Theorem.
Analysis
If f ( x)  g ( x)  h( x) in some deleted neighborhood of a
and
lim f ( x)  lim h( x)  L
x a
then
x a
lim g ( x)  L
x a
Proof
If f ( x)  g ( x)  h( x) in some deleted neighborhood of a
and
lim f ( x)  lim h( x)  L
x a
then
x a
lim g ( x)  L
x a
One-sided Limits
Common Notation
f :  b, a  
Consistency…
Limits at Infinities
Limits at Infinities
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It can be shown that (most of the) limits
laws remain valid for limits at infinities.
Example 2
Use the e-d definition to prove that
1 

lim 1  2   1
x 
 x 
Analysis
Use the e-d definition to prove that
1 

lim 1  2   1
x 
 x 
Proof
Use the e-d definition to prove that
1 

lim 1  2   1
x 
 x 
Infinite Limits
y
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y=f(x)
The left-hand limit DNE
Notation:
lim f ( x) 
x a
is not a number
a
x
Infinite Limits
Example 3
Use the e-d definition to prove that
1
lim 2  
x 0 x