It has been asked to evaluate $$\lim_{(x,y)\to(0,0)} \frac{x^3 + y^3}{x-y}$$ if it exists at all and otherwise to disprove it.
After much thoughts , I came up with an idea of substituting $y$ with $x-mx^3$ which devolved the limit to $(2/m)$ and thus served my purpose of disproving the existence of a limit.
But, thinking of such half-weird substitutions take some reasonable amount of time; the luxury of which is seldom available at examinations. So, what are other better methods to disprove the existence of this limit?
Any general algorithm (??) on disproving the existence of bi-variate limits (without indulging into trick-substitutions) will be also appreciated.
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related