How to distinguish whether $\lim {a_n\cdot b_n}$ exists or not, $\lim a_n=0$ and $\lim b_n$ doesn't exists How to distinguish whether $\lim {a_n\cdot b_n}$ exists or not, $\lim a_n=0$ and $\lim b_n$ doesn't exists.

There are following cases:
1, $\lim {a_n\cdot b_n}$ exists, however, $\lim {a_n}=0$ and $\lim {b_n}$ doesnot exist.
For example $a_n=\frac1n$ and $b_n=(-1)^n$;
2,  $\lim {a_n\cdot b_n}$ does not exist, however, $\lim {a_n}=0$ and $\lim {b_n}$ doesnot exist also .
For example $a_n=\frac1n$ and $b_n=n^2$;
