**Reading and Notes Questions**

**Question 1**. State the official definition of the sentence “ has a limit at ” that appears in our textbook.

**Question 2**. Rephrase the definition from Question 1. using the language and notation of “-balls” and “punctured -balls.”

**Question 3**. If we write , then what (if anything) can we conclude about ?

**Question 4**. On pages 65-66, our textbook explores the function . Does their discussion prove that the limit of at exists or doesn’t exist? What style of proof did they use in this discussion?

**Question 5**. In the textbook’s discussion of Example 2.5 (pages 68-69), the line “However, for a fixed positive integer , there are but a finite number of points in of the form . In fact, for a fixed positive number , there are only a finite number of points in of the form where with and positive integers.”

Explain why this line is true. Also, the function being discussed has a name, it is _______’s function. Find the name!

**Question 6**. Draw a visual interpretation of Theorem 2.1 (on page 69). Why in the statement of this theorem is it stipulated that ? In the discussion before the proof of this theorem, what previous homework problem was used?

**Question 7**. Write a very short (but rigorous!) proof of Theorem 2.2 (you may, of course, assume Theorem 2.1 is true).

**Question 8**. What is a less formal way to express the content of Theorem 2.3?

**Question 9**. Write a poem about Example 2.6.

**Question 10**. Explain exactly where Theorem 2.4 was used in Example 2.7.

**Question 11**. In the definition of “limit of a function” (on page 64), is our textbook actually claiming that is ** an interior point** for the domain of ? Explain your answer.

## Proof Questions

**Problem 1**. Prove or disprove: the set is open.

**Problem 2**. (a) Complete the following proof :

Suppose is a function with an accumulation point of , and assume and are limits of at (in accordance with the definition on page 64).

Let be given. We aim to show that , which will complete our proof.

By definition of limit, there exists a ??? > 0 so that

(assuming , too). Let such a ???? be chosen, and let and . Then

as desired. Therefore, _____ .

(b) This proof should feel suspiciously similar to the proof for a previous homework problem. Which one was/is it?

(c ) Is it possible for a function have two different limits at a single point ?

**Problem 3**. Complete exercise 11. on page 79.

**Problem 4**. Complete exercise 16. on page 80 (citing relevant theorems from Section 2.3).

**Problem 5**. Prove that any function of the form where are constants is continuous *at every point* in .

**Problem 6**. Prove that the set is neither open nor closed.