It's just one of those things, a "special limit" to basically just memorize. At least that's how I look at it.
If you want to see a proof of this particular limit, this is a good place to start:
Also, if a limit does not exist, as you posited for your k vector, that doesn't mean it's zero - zero is a number that exists just like 1, 2, 3, etc. If you find that a limit does not exist but you need to evaluate it, you need to figure out a way TO evaluate it.