My khajiit is a PURE archer. Even in melee combat I don't pull out so much as a dagger or my claws. I just slap the opponent with my bow, take a step back, and then pull back that string as fast as I can. Its a fun challenge for me.
One weakness I find with bows are guard rails. Seriously.
Seems if I'm leaning over a guard rail to shoot someone on a lower level, my arrow just sticks to the railing instead of flying to its target. I've found that I do better taking the lower ground and just shoot upwards.