Analysis of Limits using Contour Maps for the function $f(x,y)=xy/|xy|$ as $(x,y)$ $\rightarrow$ $(0,0)$ 
I'm trying to evaluate the limit as (x,y)$\rightarrow$ (0,0) of the function $z=xy/|xy|$. I have the contour plot for the function that I use to find which line I should use when deducing the limit. I know the 3d graph of z looks like 

I'm not sure along what path to check for the existence of the limit. The contour diagram doesn't show any recognizable function to me. 
In addition to the commonly used paths, y=kx and $y=kx^2$, what are some other paths I can try to find the multivariable limit? The answer keys says that I should use the path y=kx and I don't see why I should use the path y=kx rather than some other path such as $y=kx^2$ or y=1/x? or y=sin(x) or y=e^x or y=tan(x).
Side question, if appropriate, can we use any function as a path that we use in the two-path test for the existence of a limit?


If I had the two contour plots shown above for some function, what path would I use to test for the existence of a limit? I don't think y=kx or y=kx^2 fit the sequence shown in the contour plot. 
 A: If you find two different limits, following two different paths respactively, then the limit does not exist.
It is proved that if a limit of a function exists at a given point then it is unique.
Thus uniqueness of the limit is a necessary condition for the existence of the limit.
For $\frac{xy}{|xy|}$ if you take the path $(x,x)$ i.e $x=y$ then you have that 
$\frac{xy}{|xy|}=\frac{x^2}{|x^2|}=1 \rightarrow 0$ as $x \rightarrow 0$
If you take the path $(x,-x)$ ,  then $$\frac{xy}{|xy|}=\frac{-x^2}{|x^2|}= -1 \rightarrow -1$$  as $x \rightarrow 0$
Thus we found two paths that go to zero and the limit in the fisrt path is $1$
and in the second path is $-1$
If the limit existed at $(0,0)$ then the value of it, would independent of our choices of paths.
In simpler words it would have tha same value on all paths we choose to approximate it.
If you want a second way the you can do this:
Take polar coordinates: $$x=r \cos{t}$$ $$y=r \sin{t}$$
If you substitute you will have the expression $$\frac{\cos{t} \sin{t}}{|\cos{t} \sin{t}|}$$
If thelimit exists then its value would be independent of $t$ as $r \rightarrow 0$
But for $t_1= \frac{\pi}{3}$ and $t_2=\frac{\pi}{4}$ you have differend values for the above expression.Thus the limit does not exist.
