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

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

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