is ambiguous because the limit depends on what path you take to approach (0,0). For example, if you approach from the path y = x, you would obtain 1/2, but if you went from the path y = 0, the limit would be zero. Therefore the limit doesn't exist.

As for the first limit, I haven't studied complex limits but still, you know that z is a complex number but you have no idea which direction z is coming from in order to approach (0,0) or 0 + 0i. However, you know that

and

so

Of course, theta is dependent on which path you are taking...if you are traveling on a straight line path towards the origin (e.g. ), then theta would be constant and the limit would just be