Find the limit of the function $f(x,y)=\frac{x^2 y^2}{x^2 y^2 + > (x-y)^2}$ whenever $x^2 y^2 + (x-y)^2 \not= 0$ as $(x,y) \to (0,0) $
I thought I can split this question into 4 limit question like $$\lim_{(x,y) \to (0^+, 0^+)} f(x,y)$$ $$\lim_{(x,y) \to (0^+, 0^-)} f(x,y)$$ $$\lim_{(x,y) \to (0^-, 0^+)} f(x,y)$$ $$\lim_{(x,y) \to (0^-, 0^-)} f(x,y)$$
Logically, this way of looking the question should not be a problem, but it is to a efficient way to do it, especially if f(x,y) is more complicated than the one in this question.
So, how can I solve this question more efficiently ? and what is the correct solution ?
The limit can be split into two cases.
$1.$ The limit is traversed along the line $x=y$
$2.$ The limit is not traversed along the line $x=y$
Case $1$: Substitution leads to direct cancellation, and the limit in that case is $1$.
Case $2$: By setting $y=kx$ where $k \neq 1$, the limit is $0$.
Conclusion: The limit does not exist.