I'm not sure if you have to use delta-epsilon or not, but you can use the fact that 2*sin(theta)*cos(theta) = sin(2*theta) and show that if theta doesn't converge then neither does the limit.
You will have to also partition (-1)^n * n with the other term as well, but the above is my suggestion to showing that it doesn't exist (you can show that (-1)^n also doesn't exist by writing it as a trig term of cos(pi*n) = (-1)^n).
I have some ideas to get you started . . .
We have: .
The first limit fluctuates between +1 and -1.
The second limit is the sine of a large multiple of
. . . .
The third limit is 1: .
Can you continue?