This is my first post on Stack Exchange, if I am violating any posting rules, I will gladly edit my post according to the required parameters.
The question I have says for (x, y) $\neq$ (0, 0) $$f(x, y) = \frac{x^3}{x^3 + y^2 - 2x}$$ Show that for f(0, 0) = 0, f(x, y) is not continuous.
Using double limit and the substitution $my^2 = x^3$, I seem to be able to show the function is continuous.
The only way to show the limit does not equal zero seems to be a $my^2 = x^3 - 2x$ substitution and then using double limit again OR using a $mx^3 = y^2 - 2x$ substitution and then cancelling the $x^3$ in the numerator and denominator. Both these methods give different limits, but both having the arbitrary constant "m".
However, both these methods seem sketch to me on different grounds, as in the first one I still have 2 variables after substitution and in the second method, I do not have all variables on one side of the equals sign. I do not know the significance of these petty discomforts in substitution.
Something I have tried doing is changing the subject of the substitution from y to x in the equation $$my^2 = x^3 - 2x$$
However, this has proved to be more difficult than I originally expected. I still think changing the subject in the above equation would give a conclusive answer to what the limit is.