General Sequence Limits Question and Method Know How

If searched the forums for the explanation of the famous

- $\displaystyle \lim_{n\to\infty}\frac{sin(n)}{n}$

I am not saying the explainations in some thread weren't concrete, its just something about this limit or I am just over complicating a simple things here, I guess teachers like to use this example because the $\displaystyle sin(n)$ seems to throw people off and this limit infact behaves similarily to just $\displaystyle \frac{1}{n}$ It seems like I am just complicating things that are obvious, please feel free not to waste your time and link me to an explanation if you'd like.

Also my other question, how do you know what theorem or tests works the best for the limits or convergence of a sequence function, as in what are the mains things you look for?