Project: The Significance of the “For Every”

Section 2.4 - 15 points

Due:

Consider the following functions: and

We want to try to prove the following statements:

Notice that these are not “obvious statements, since g(2) is undefined.

1. We start with .

a) Can you find a number with the property that, when ?

Illustrate your answer with a graph.

b) Can you find a number with the property that, when ?

2. We now have some reason to believe that the above statements are true. But just having

“some reason to believe” isn’t enough for mathematicians. Repeat the previous problem

with

3. Based upon the above work, could you find a for every that could be chosen.

What does this tell you about the limits?

Infinity is very Big

1. For what values of x near 0 is it true that ?

2. For what values of x near 0 is it true that ?

3. For what values of x near 0 is it true that ?

4. What do the above statements say about the function for values very near 0?

5. Why is it not true that ? Give reasons for your answer.