Belt is fairly easy to check on the N47 engines as it's right at the front although on the F30 the air filter at the front gets in the way and there is a cover over it. You'll need to unscrew and pull that back then get a torch and have a look at it as best you can to make sure there's no splits or cracks in the rubber and that it just generally looks alright. Here's a random pic to show what I mean:
They're dead easy to change on these engines - not had to do it on my F30 but I did it on my E90 with N47 and it just needed a socket wrench to hold and release the tensioner then a drill bit to hold it in place. The belt then just comes off and the new one slips on, access was not a problem and it was less than a 10 minute job all in all. As mentioned before it's a bit different on the F30 N47 as the air filter housing and a cover is in the way but should still be easy enough even with removing that.
Tensioner would usually be making some kind of noise if it needed changing, a squeal or some other kind of noise. If you can't hear any strange noises from the belt area at the front of the engine you're probably fine.