Prove that $\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$ does not exist

Question

Given $\frac{a}{c} + \frac{b}{d} = 1$ Prove that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$$ does not exist.

So I have done the proof for strict inequalities. And the limit only exists when the fractions add up to an integer greater than $1$. I'm not sure how to approach this other than counter examples.

Answer 1

1. Taking the limit along the axes gives zero.
2. Taking $|x|^c=|y|^d=t\to 0$ gives $$ \frac{t^\frac{a}{c}\cdot t^\frac{b}{d}}{t+t}=\frac{t}{2t}=\frac12. $$