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I can't prove or disprove the existence of $\lim_{(x,y)\to (0,0)}\frac{x^3+y^2}{x^2+y}$. I have already tried several paths but still I can't prove that the limit doesn't exist. Any ideas?

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    $\begingroup$ Try a quadratic path. This function isn't even defined in a neighborhood of the origin.... $\endgroup$
    – user296602
    Apr 24 '18 at 1:56
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    $\begingroup$ You have a big problem if $y+x^2=0$. $\endgroup$ Apr 24 '18 at 2:01
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Let $y=x^{4}-x^{2}$, then \begin{align*} \lim_{x\rightarrow 0^{+}}\dfrac{x^{3}+(x^{4}-x^{2})^{2}}{x^{2}+x^{4}-x^{2}}=\lim_{x\rightarrow 0^{+}}\dfrac{x^{8}-2x^{6}+x^{4}+x^{3}}{x^{4}}=\infty. \end{align*}

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