Understanding the Squeeze Theorem in Analysis

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.

• Squeeze Theorem
• Analysis
• Limits
• Calculus
• Proof

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1. MAT 3749 Introduction to Analysis Section 2.1 Part 3 Squeeze Theorem and Infinite Limits http://myhome.spu.edu/lauw

2. 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

3. References Section 2.1

4. 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

5. Lemma 2 (Expanded) 2 2 0 0 k or x k x k x k 0, k 2 2 x k x k

6. Preview Squeeze Theorem One-sided Limits Limits at Infinities Infinite Limits

7. 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

8. Squeeze Theorem ? ( ) f x ( ) g x ( ) h x (x ) h (x ) g (x ) f ? ?

9. Squeeze Theorem ? ( ) f x ( ) g x ( ) h x (x ) h = = lim ( ) x a lim ( ) x a f x h x L ? (x ) f ? ?

10. 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 ? ?

11. Squeeze Theorem ? (x ) h You will see this type of idea over and over again. (x ) g ? (x ) f ? ?

12. Example 1 ( ) 1 x 1 x 2 2 lim x sin lim x limsin x x x 0 0 0

13. Example 1 ( ) 1 x 1 x 2 2 lim x sin lim x limsin x x x 0 0 0

14. 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)

15. 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.

16. 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

17. 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

18. One-sided Limits

19. Common Notation ( ) b a : , f

20. Consistency

21. Limits at Infinities

22. Limits at Infinities It can be shown that (most of the) limits laws remain valid for limits at infinities.

23. Example 2 Use the e-d definition to prove that 1 x = lim 1 x 1 2

24. Analysis Use the e-d definition to prove that 1 x = lim 1 x 1 2

25. Proof Use the e-d definition to prove that 1 x = lim 1 x 1 2

26. Infinite Limits y The left-hand limit DNE Notation: y=f(x) = lim x ( ) f x a is not a number x a

27. Infinite Limits

28. Lemma 2 (Expanded) 2 2 0 0 k or x k x k x k 0, k 2 2 x k x k

29. Example 3 Use the e-d definition to prove that 1 x = lim x 2 0