Principles of Analysis

Homework Assignment #7

Due Wednesday, Oct. 30, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to Bilodeau and Thie's book, the required text for the course. You should write up solutions neatly to all problems, paying particular attention to your arguments and proofs. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.

Section 3.2
Problems:   2a, 6

Hint: For problem #2a, start by reading the proof of the first part of Thm. 3.2.3 (the BLT for functions). Amend the proof in the text to the subtraction case.

Section 3.3
Problems:   1 (parts d, e, f), 3 (parts a, b, c), 5 (parts c, d, e)

Hint: The only problems requiring rigorous proofs are those in #5. The proofs are similar to those you have been doing in Section 3.1. Use class notes and the text for specific examples.

Section 3.4
Problems:   1 (parts b, g, l), 11 (parts a, b)

Hint: When computing the necessary limits in problem #1, try rearranging the expression to make use of limits we already know. For example, in 1a (not assigned) you could multiply top and bottom of the fraction by 2, and write a(x) = 2 times (sin 2x)/(2x). We then can apply the Big Limit Theorem to conclude lim x -> 0 of a(x) = 2.

Extra Problem: Prove the Squeeze Theorem for functions. See your class notes from 10/23 for a precise description of the theorem and for hints.