Prove that if f(x) is defined as f(x)=1 for x>0 and f(x)=-1 for x<0, then the limit as x approaches zero does not exist.
We use a proof by contradiction.
Suppose that `lim_(x->0)=L` , where `L in RR` . Then for all `epsilon>0` there exists a `delta` such that `0<|x|<delta => |f(x)-L|<epsilon` .
We let `epsilon = 1/3` .(Any positive value less than 1/2 will do). Then for x>0 we have `-1/3<1-L<1/3` or `2/3<L<4/3` . For x<0 we have `-1/3<-1-L<1/3` or `-4/3<L<-2/3` .
This is impossible; no real L lies in both of these intervals simultaneously thus contradicting the existence of the limit.
Therefore, the limit does not exist.