I managed to solve the first problem, which I removed from the first post for neatness/lack of clunkiness. I also made progress on the second problem, though I'm not sure if its in the correct direction or not, and I'm hoping I'll get some help here. I really want to understand this, so when you correct me, please don't just give me an answer but also explain and elaborate on where I went wrong in my work so I can keep it for the record.
a) Here is the conclusion I reached. For all x with 0 < | x - a | < δ, we have | f(x) - f(a) | < ε.
In this case, f(x) = 6, and f(a) = 4. Therefore, would it be right to say that | 6 - 4 | < ε, or | 2 | < ε? That would mean there is no such value for δ, since ε = 2, and | 2 | is not greater than 2.
b) For this second part of the problem, here is what I came up with. Again, the same rule applies:
For all x with 0 < | x - a | < δ, we have | f(x) - f(a) | < ε. f(x) is still 6, and f(a) is still 4. However, the value of ε has changed; now ε = 1. Thus, here is what I did: | 6 - 4 | < ε, or | 2 | < ε. This would also mean that there is no such value for δ since ε = 1, and 1 is NOT greater than 2.
c) For c, I am not sure how to proceed, and would like some sort of nudge in the right direction. A guess here though is that if none of the two above possibilities worked, wouldn't this one too mean that there is no relationship between δ and ε?