Any necessary and sufficient condition for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$? I am looking for any established result that is necessary and sufficient for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$ for any real sequences $\{a_n\} \text{and} \{b_n\}\ne 0$, where both the two sequences converges to the same limit.

Can any one help me with some hints comments or suggestions?
 A: Necessary/sufficient conditions type of problems can sometimes seem hard to start, but there are techniques to easily approach them.
One way is to start with a necessary condition (respectively, a sufficient condition), and improve on it until you feel your necessary condition becomes sufficient (respectively, until your sufficient condition becomes also necessary), and then try to prove that it is in fact sufficient (resp. necessary).
For example,
as Robin Saunders commented,
if $a_n\to a$ and $b_n\to b$ with $a,b$ nonzero and not $\pm\infty$,
then the quotient rule for limits (see Wikipedia) implies that the radio $a_n/b_n$ will converge to $a/b\neq0$.
Thus,
a first necessary condition is the following:

At least one of $a_n$ or $b_n$ converges to $0$ or $\pm\infty$.

Is this condition also sufficient? Clearly not,
as $a_n=n^2$ and $b_n=n$ are both such that $a_n,b_n\to\infty$,
so the necessary condition is satisfied,
yet $a_n/b_n$ converges to + infinity!
So the work is not done: you have to make your necessary condition stronger, i.e., dismiss other cases that won't work. However, you're not as far as you were when you started, because now you only have to consider the cases where at least one of $a_n$ or $b_n$ converges to $0$ or $\pm\infty$.
