The limit I need to calculate is $\lim_{(x,y)\rightarrow (0,0)}\frac{xy^{2}}{x^{4}+y^{2}}$. Using polar coordinates I get: $lim_{r\rightarrow 0}\frac{r\cos(\theta)\sin^{2}(\theta)}{r^{2}\cos^{4}(\theta)+\sin^{2}(\theta)}$. Now if $\sin(\theta)\neq 0$ then the limit is $0$. How do I handle the case where $\theta=0$ or $\theta = \pi$? And is there a better way to approach this limit?
limit with two variables
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limits
multivariable-calculus