Questions about existence of limit of a function....? If  f : R--->R be defined by:   f(x)={ 1  when x>0  and  -1  when x<0 Please prove that limit as x-->0 does not exist, with epsilon--delta definition...?

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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.

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