How to prove this limit does not exist? I have the following limit: 

$$\lim_{(x,y) \to (0,0)}  \frac{\sin(xy)}{\sqrt{x^2 + y^2 + |y|} \times \|(x,y)\|}$$

Where I'm using the usual euclidean norm. I've been trying to find a curve that approximates (0,0) in a way such that the limit is not 0, but I'm really struggling to do so.
We have that

$$\lim_{(x,y) \to (0,0)}  \frac{\sin(xy)}{\sqrt{x^2 + y^2 + |y|} \times ||(x,y)||}=\lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy} \frac{xy}{\sqrt{x^2 + y^2 + |y|} \times ||(x,y)||}$$

and

$$\lim_{(x,y) \to (0,0)}  \frac{\sin(xy)}{xy}=1$$

and by polar coordinates 

$$\frac{xy}{\sqrt{x^2 + y^2 + |y|} \times ||(x,y)||}=\frac{r\cos \theta \sin \theta}{\sqrt{r^2+r|\sin \theta|}}\to 0$$

indeed 

$$0\le \frac{r|\cos \theta| |\sin \theta|}{\sqrt{r^2+r|\sin \theta|}}\le \frac{r|\cos \theta| |\sin \theta|}{\sqrt{r|\sin \theta|}} \frac{\sqrt{r|\sin \theta|}}{\sqrt{r|\sin \theta|}}=|\cos \theta|\sqrt{r|\sin \theta}| \to 0$$
Let $x=rc,y=rs$ with $c^2+s^2=1$ and $r$ small. Then the numerator is on the order of $r^2cs$ and the denominator on the order of $\sqrt{r|s|}$ (because $r^2$ is neglectible, unless $s=0$) times $r$. Hence the limit converges to zero like $\sqrt r$.

If we can't neglect $r^2$ because $s$ tends to zero,
$$\frac{r^2cs}{r\sqrt{r^2+r|s|}}=\frac{cs}{\sqrt{1+\dfrac{|s|}r}}$$ tends to zero anyway.