This is usually proved not by definition, but from geometric arguments. See PlanetMath, here, at Dr. Math Forum with some pretty bad ASCII notation. You can also Google for "sin(x) / x".
Yes, though this does not move you very far.I have abs abs((sin x / x) -1)< epsilon and since sin x / x will always be less then 1 it would just become (1- sin x / x) < epsilon right?