Is $(\frac 1{n^2 \sin n })$ convergent ? If so , what is the limit? Is the sequence  $\left(\dfrac 1{n^2 \sin n }\right)$ convergent ? If it does, with what limit?
If it converges to a not 0 then from 1/n^2->0 and by division we get 1/sinn-->0 or sinn-->infinity which is absurd.
If it converges to 0 then 1/(nsinn) -->0 and so n.sinn-->infinity. Thus there is N such that n>N implies that n.sinn is positive. This is absurd too,
The sequence does not converge.
If it converges to a not 0 then from 1/n^2->0 and by division we get 1/sinn-->0 or sinn-->infinity which is absurd.
If it converges to 0 then 1/(nsinn) -->0 and so n.sinn-->infinity. Thus there is N such that n>N implies that n.sinn is positive. This is absurd too,
The sequence does not converge.