Let's look at (iii). I claim the limit is zero: that is, for all epsilon > 0 there exists delta > 0 such that if 0 < sqrt(x^2+y^2) < epsilon then | xy(|x|+|y|) | < delta.

I claim we can take delta = epsilon. If 0 < sqrt(x^2+y^2)<epsilon, then |x| and |y| are less than epsilon and one of them, say x, is non-zero. Then |x|.|y|/(|x|+|y| < epsilon. |x|/(|x| + epsilon) < epsilon as claimed.