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

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

x y a

x y L a

x y L a

You will see this type of idea over and over again. x y L a

Example 1

We cannot apply the limit laws since DNE (2.1.1)

Example 1 Make sure to quote the name of the Squeeze Theorem.

Analysis

Proof

One-sided Limits

Common Notation

Consistency…

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 y a y=f(x) The left-hand limit DNE Notation: is not a number

Infinite Limits

Example 3 Use the e-d definition to prove that

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