This is a very beginner problem that I've had ever since this concept was introduced in our class. Basically one of the methods to solve limits that was presented to us involved setting a variable equal to a "curve" (which preferably cancelled out some other nasty parts of the function) and evaluate the limit, and then do the same with another curve. If the evaluated limits differed, one could then conclude that it didn't exist.
Is this something you can just do with no restrictions? Does it suffice to just "plot" two curves and see if you get the same result? This seems way too easy or too risky because of the way you could manipulate the functions. Maybe you're restricted to just using level curves...? I'm very lost.
Here is an example where we did something like I attempted to explain above:
My first try is to apply a linear relationship between the variables.
$y=kx$ and $z=mx$.
Turn the multivariate limit to a univariate limit and try to calculate the limit. If the limit depends on $m,k$ you can conclude the limit does not exist.
Note that if the limit does not depend on $m,k$ this does not imply that the limit exists.