Squeeze Theorem in Analysis
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MAT 3749
Introduction to Analysis
Section 2.1 Part 3
Squeeze Theorem and
Infinite Limits
http://myhome.spu.edu/lauw
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
Recall 0.1.3 Lemma 2
Lemma 2 (Expanded)
Preview
Squeeze Theorem
One-sided Limits
Limits at Infinities
Infinite Limits
Squeeze Theorem
Squeeze Theorem
Squeeze Theorem
Squeeze Theorem
Squeeze Theorem
You will see
this type of
idea over and
over again.
Example 1
Example 1
Example 1
We cannot apply the limit laws since
DNE (2.1.1)
Example 1
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Analysis
Proof
One-sided Limits
Common Notation
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
Analysis
Use the
e
-
d
definition to prove that
Proof
Use the
e
-
d
definition to prove that
Infinite Limits
x
The
left-hand limit
DNE
Notation:
is not a number
Infinite Limits
Lemma 2 (Expanded)
Example 3
Use the
e
-
d
definition to prove that
Explore the Squeeze Theorem and its applications in infinite limits, one-sided limits, and limits at infinities. Discover the core concepts and examples to grasp the importance of this theorem in analysis and calculus.
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Presentation Transcript
MAT 3749 Introduction to Analysis Section 2.1 Part 3 Squeeze Theorem and Infinite Limits http://myhome.spu.edu/lauw
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
Recall 0.1.3 Lemma 2 2 2 0 0 k or x k x k x k 0, k 2 2 x k x k
Lemma 2 (Expanded) 2 2 0 0 k or x k x k x k 0, k 2 2 x k x k
Preview Squeeze Theorem One-sided Limits Limits at Infinities Infinite Limits
Squeeze Theorem If ( ) and lim ( ) x ( ) g x ( ) in some deleted neighborhood of lim ( ) x a g x L = f x h x = a = f x h x L a then lim ( ) x a
Squeeze Theorem ? ( ) f x ( ) g x ( ) h x (x ) h (x ) g (x ) f ? ?
Squeeze Theorem ? ( ) f x ( ) g x ( ) h x (x ) h = = lim ( ) x a lim ( ) x a f x h x L ? (x ) f ? ?
Squeeze Theorem ? ( ) f x ( ) g x ( ) h x (x ) h = = lim ( ) x a lim ( ) x a f x h x L (x ) g = ? lim ( ) x ag x L (x ) f ? ?
Squeeze Theorem ? (x ) h You will see this type of idea over and over again. (x ) g ? (x ) f ? ?
Example 1 ( ) 1 x 1 x 2 2 lim x sin lim x limsin x x x 0 0 0
Example 1 ( ) 1 x 1 x 2 2 lim x sin lim x limsin x x x 0 0 0
Example 1 ( ) 1 x 1 x 2 2 lim x sin lim x limsin x x x 0 0 0 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 x = = lim ( ) x a lim ( ) x a f x h x L sin = lim ( ) x ag x L Make sure to quote the name of the Squeeze Make sure to quote the name of the Squeeze Theorem. Theorem.
Analysis If ( ) and lim ( ) x ( ) g x ( ) in some deleted neighborhood of lim ( ) x a g x L = f x h x = a = f x h x L a then lim ( ) x a
Proof If ( ) and lim ( ) x ( ) g x ( ) in some deleted neighborhood of lim ( ) x a g x L = f x h x = a = f x h x L a then lim ( ) x a
Common Notation ( ) b a : , f
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 1 x = lim 1 x 1 2
Analysis Use the e-d definition to prove that 1 x = lim 1 x 1 2
Proof Use the e-d definition to prove that 1 x = lim 1 x 1 2
Infinite Limits y The left-hand limit DNE Notation: y=f(x) = lim x ( ) f x a is not a number x a
Lemma 2 (Expanded) 2 2 0 0 k or x k x k x k 0, k 2 2 x k x k
Example 3 Use the e-d definition to prove that 1 x = lim x 2 0
