How do I prove limx^2 as x approaches c is c^2

So far I have
(limx^2) x--->c = c^2 because

If c^2 = L

|x^2 - c^2 | < epsilon

| x - c | * | x + c | < epsilon

| x - c | < epsilon/| x + c |

The obscure trick with these is to make an assumption that |x - c| is sufficiently small, and then force that assumption to be true with your choice of delta. This controls the |x + c| factor, and makes it manageable. However, it's easier to do when we know what the value of c is. As it stands, we will have to prove in three cases:

1) c > 0
2) c < 0
3) c = 0

Case 3) is the easiest, and I think you'll have no trouble doing it. Let's try case 1). We need to make an assumption, specifically that |x - c| < c/2. Remember that c > 0 ===> c/2 > 0. We want to turn this statement into a statement regarding x + c:

|x - c| < c/2
===> -c/2 < x - c < c/2
===> 0 < 3c/2 < x + c < 5c/2
===> |x + c| > 3c/2

This is important, because now:

epsilon / |x + c| < epsilon / (3c/2) = 2epsilon / (3c)

So, if we choose delta to be the minimum of 2epsilon / (3c) and c/2, then we force |x - c| < c/2, as per our assumption, as well as forcing (by our assumption) |x - c| to be less than epsilon / |x + c|.

Try case 2) yourself.