# Determining whether $f(x,y)$ is continuous at $(0,0)$

by iamokay   Last Updated January 16, 2018 07:20 AM

$f(x,y)=\begin{cases} \dfrac{x^2y^{\frac{4}{3}}}{x^4+y^2}&(x,y)\neq(0,0)\\ 0&(x,y)=(0,0) \end{cases}$

How do I determine whether $f$ is continuous at $(0,0)$? That is to say, is $$\lim_{(x,y)\to(0,0)}f(x,y)$$ equal to $0$?

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$(x^2-|y|)^2 \ge 0$ gives $x^4+y^2 \ge 2 x^2 |y|$, so ${|x^2 y| \over x^4+y^2} \le {1 \over 2}$.
Hence ${|x^2 y^{4 \over 3}| \over x^4+y^2} \le {1 \over 2}|y|^{1 \over 3}$.