Math task 10 hrs

Name: Math 325 BRONZE Assignment # 19

Assigned: 2020.04.06 Due: 2020.04.10

Recall that you may be asked to present your solutions to Bronze questions in class and that Bronze Questions are to be turned in at the end of class.

For Friday, April 10 (Note the slightly extended due date.)

• Get 7–9 hours of good sleep each night. Sleep is the basis for memory and creative thought, so your sleep time should be as regular as possible and absolutely non- negotiable. A cold, pitch-black environment, with absolutely no blue light can help. More sleep tips can be found online.

Note: Losing even an hour of sleep can strongly impact your immune system’s performance.

I Read Sections 6.1, 6.2, and 6.3 in Cummings.

• Bronze Questions

1. (Exercise 6.1(a) in Cummings) Use the �-δ definition of the functional limit to prove that limx→−2(4x+ 3) = −5.

2. (Exercise 6.1(b) in Cummings) Use the �-δ definition of the functional limit to prove that

limx→1 x3−1 x−1 = 3.

3. (Exercise 6.1(c) in Cummings) Use the �-δ definition of the functional limit to prove that limx→0 x

2 = 0.

4. (Exercise 6.1(d) in Cummings) Use the �-δ definition of the functional limit to prove that limx→2 x

3 = 8.

5. (Exercise 6.4(a) in Cummings) If limx→a f(x) and limx→a g(x) both do not exist, can limx→a[f(x) + g(x)] exist? Prove your result, or give a counterexample.

6. (Exercise 6.11 in Cummings) Suppose that f : X → Y , and that {Bα}α∈I is a (possibly uncountable) collection of subsets of Y . Prove that the pre-image of the union is the union of the pre-images, that is, prove that

f−1

(⋃ α∈I

Bα

) = ⋃ α∈I

f−1 (Bα) .