<s>[INST] $\lim\limits_{(x,y)\rightarrow(0,0)} \frac{7 \sin(2 x) x^2 y}{2 x^3 + 2 x y^2}$ Does the following limit exist?
$$\lim_{(x,y)\rightarrow(0,0)}  \frac{7 \sin(2 x) x^2 y}{2 x^3 + 2 x y^2}$$
I've shown that it exist but i'm unsure about how to find the value of the limit.
 [/INST] Note that
$$\frac{7 \sin(2 x) x^2 y}{2 x^3 + 2 x y^2}=7\cdot\frac{\sin(2x)}{2x}\cdot\frac{x^2y}{x^2+y^2}.$$
When $(x,y)\to (0,0)$, we have $x\to 0$ and $r=\sqrt{x^2+y^2}\to 0$, therefore, we have 
$$\left|\frac{\sin(2x)}{2x}\right|\to 1\mbox{ as }x\to 0$$
and 
$$\left|\frac{x^2y}{x^2+y^2}\right|=\left|\frac{r^3\cos^2\theta\sin\theta}{r^2}\right|\leq r\to 0.$$
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