MAT 3749 Introduction to Analysis Section 2.3 Part I Derivatives .
MAT 3749 Introduction to Analysis Section 2.1 Part 3 Squeeze Theorem and Infinite Limits .
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MAT 3749 Introduction to Analysis Section 2.1 Part 3 Squeeze Theorem and Infinite Limits http://myhome.spu.edu/lauw
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Transcript of MAT 3749 Introduction to Analysis Section 2.1 Part 3 Squeeze Theorem and Infinite Limits .
MAT 3749Introduction to Analysis
Section 2.1 Part 3
Squeeze Theorem and Infinite Limits
http://myhome.spu.edu/lauw
Notes
Group reassignments Math Party Exam 1 Please study for the quizzes
Major Themes
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
Section 2.1
Preview
Squeeze Theorem One-sided Limits Limits at Infinities Infinite Limits
Squeeze Theorem
If ( ) ( ) ( ) in some deleted neighborhood of
and lim ( ) lim ( )
then lim ( )x a x a
x a
f x g x h x a
f x h x L
g x L
Squeeze Theorem
( ) ( ) ( )f x g x h x
x
y
a
)(xf
)(xg
)(xh
Squeeze Theorem
x
y
L
a
)(xf
)(xh( ) ( ) ( )f x g x h x
lim ( ) lim ( )x a x a
f x h x L
Squeeze Theorem
x
y
L
a
)(xf
)(xg
)(xh( ) ( ) ( )f x g x h x
lim ( ) lim ( )x a x a
f x h x L
lim ( )x ag x L
Squeeze Theorem
You will see
this type of
idea over and
over again.
x
y
L
a
)(xf
)(xg
)(xh
2 2
0 0 0
1 1lim sin lim limsinx x xx x
x x
Example 1
Example 1
2 2
0 0 0
1 1lim sin lim limsinx x xx x
x x
Example 1
We cannot apply the limit laws since
DNE (2.1.1)
xx
1sinlim
0
2 2
0 0 0
1 1lim sin lim limsinx x xx x
x x
Example 1
( ) ( ) ( )f x g x h x
lim ( ) lim ( )x a x a
f x h x L
lim ( )x ag x L
Make sure to quote the name of the Squeeze Theorem.
1sin
x
Analysis
If ( ) ( ) ( ) in some deleted neighborhood of
and lim ( ) lim ( )
then lim ( )x a x a
x a
f x g x h x a
f x h x L
g x L
Proof
If ( ) ( ) ( ) in some deleted neighborhood of
and lim ( ) lim ( )
then lim ( )x a x a
x a
f x g x h x a
f x h x L
g x L
One-sided Limits
Common Notation
: ,f b a
Consistency…
Limits at Infinities
Limits at Infinities
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
2
1lim 1 1x x
Analysis
Use the e-d definition to prove that
2
1lim 1 1x x
Proof
Use the e-d definition to prove that
2
1lim 1 1x x
Infinite Limits
x
y
a
y=f(x)
lim ( )x a
f x
The left-hand limit DNE Notation:
is not a number
Infinite Limits
Example 3
Use the e-d definition to prove that
20
1limx x
