Weird Limit

I guess its new puzzle time. I do have a new puzzle to post, but I haven't fully solved it out yet, and I don't like posting puzzles I don't have full solutions to, since I'm worried it will take longer than I expect. Instead I'm going to post a limit I saw a few years ago that I thought was really neat. Thats right, its time for math. Anyway,

lim_N->∞ N sin(N!2πe)

where the limit of N is taken along integer values (which is why I can use factorial instead of gamma or something). Its interesting as the coefficient out front is just going to diverge linearly, and the argument of the sin function is just seemingly random numbers, so it should be random numbers times a huge number, its actually amazing that the limit exists at all.

Anyway, the limit does exist as a nonzero real number, I'll show it next time, but it turns out it converges by the magic of e. Well, not quite just e, if you replaced e by e+r with any rational number r, this limit would still exist (that is a bit of a hint for you).