<s>[INST] Does the limit exist? (Calculus) Consider the function 

$$f(x,y)=\frac{2xy^2\sin^2(y)}{(x^2+y^2)^2}.$$ 

Does the limit exist when $(x,y)$ tends to $(0,0)$?
 [/INST] Just see this

$$ \frac{2xy^2 \sin^2(y)}{(x^2+y^2)^2}  \sim \frac{2 x y^4}{(x^2+y^2)^2} $$

since $\sin t \sim_{t\sim 0} t$. Then we have
$$ \bigg|  \frac{2 x y^4}{(x^2+y^2)^2}  \bigg| \leq   \frac{2 |x| |y|^4}{(x^2+y^2)^2} \leq \frac{2\sqrt{x^2+y^2}  (\sqrt{x^2+y^2})^4}{(x^2+y^2)^2} = 2\sqrt{x^2+y^2} < \epsilon $$
$$ \implies \sqrt{x^2+y^2} < \frac{\epsilon}{2} =\delta. $$
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