Well, you are looking at a subset of the interval . What do you know about in that interval? Where is it zero? Where is it one? Where is it somewhere in between? Let's try some points...

. Therefore, . What other points are in there? For what points will that limit not exist? For instance, what is ? I guess that depends on if infinity is periodic (hint: it usually is NOT. Unless it is periodic for all finite examples, then it will NOT be periodic in the limit.) Figure out when sin is constant for all increasing integers (like we did for 0), and you will find when its limit exists.