How can I prove that the limit doesn't exist? I have to study the following limit
$$\lim_{(x,y)\to(0,0)}\frac{1-\cos(\sqrt{|xy|})}{x}.$$
I think that this limit does not exist, so I'm trying to prove it. First, I discovered that, if $x=y$, then, the limit is equal to zero. Is there any other variable changing that I can use?

hint
The limit exists and it is zero.
$$(\forall x,y\in \Bbb R)\;\;$$
$$ 1-\cos(\sqrt{|xy|})=2\sin^2(\frac{\sqrt{|xy|}}{2})$$
$$\le \frac 12|xy|$$
because
$$(\forall X\in \Bbb R)\;\;|\sin(X)|\le |X|$$
thus, if $ x\ne 0$,
$$|f(x,y)|\le \frac 12|y|$$
