<s>[INST] Find $\lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2}$ 
$$\lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2}$$

how should I approach this?
$$\lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)}=0$$ but I still have $$\lim_{(x,y)\to(0,0)}\frac{1}{x^2y^2}$$
The answer says there is no limit
 [/INST] Note that
$$\lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}=\frac{1}{2}.$$
So in there is a neighborhood $U$ of $(0,0)$ where $\frac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}\geq \frac{1}{4}.$
Hence, for $x\not=0$, $y\not=0$ and $(x,y)\in U$,
$$\frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2}=\frac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}\cdot \frac{x^2+y^2}{x^2y^2}\geq \frac{1}{4}\cdot \left(\frac{1}{x^2}+\frac{1}{y^2}\right)\to +\infty$$
as $(x,y)\to(0,0)$. 
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