prove limit doesn't exist $\lim\limits_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2}$ I need to show that limit doesn't exist:
$\lim\limits_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{(x^2+y^2)x^2y^2}$
How can I show it? 

First, not that for $xy\ne 0$, we have
$$\begin{align}
\frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)}&=\frac{2\sin^2(x^2+y^2)}{x^2y^2(x^2+y^2)}\\\\
&=2\left(\frac{\sin(x^2+y^2)}{x^2+y^2}\right)^2\times \left(\frac{x^2+y^2}{x^2y^2}\right)\tag1
\end{align}$$
The first term on the right-hand side of $(1)$, $\left(\frac{\sin(x^2+y^2)}{x^2+y^2}\right)^2$, tends to $1$ as $(x,y)\to(0,0)$ However, the second term, $\frac{x^2+y^2}{x^2y^2}$ tends to $+\infty$ since
$$\frac{x^2+y^2}{x^2+y^2}\ge \frac{1}{x^2}\to \infty$$
Therefore, we see that 
$$\lim_{(x,y)\to(0,0)}\frac{1-\cos(x^2+y^2)}{x^2y^2(x^2+y^2)}=\infty$$
