No. If the limit exists it is unique. CaptainBlankgave a nice proof of this.

No consider the classic example (-1)^n2.A sequence {an} that has no limit diverges

It has no limit.3.the sequence {1,-1,1,-1..}has two limits.

Yes. But it does not work the other way around always, so you need to be careful.4.If lim n->inf {an}=L and Lim n-> inf{bn}=M then lim n->inf {an bn } =LM

No, it can be shown two functions are equal only if they have the same domain and for each x in D we have f(x)=g(x).5 f(x)=(x^2-1)/(x-1) and g(x) =x+1 are same function

The domain of f(x) doth not include 1.

But g(x) does.

Equality is wrong.