TITLE: Prove that $\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$ does not exist
QUESTION [8 upvotes]: 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.

REPLY [5 votes]: Taking the limit along the axes gives zero.
Taking $|x|^c=|y|^d=t\to 0$ gives
$$
\frac{t^\frac{a}{c}\cdot t^\frac{b}{d}}{t+t}=\frac{t}{2t}=\frac12.
$$