How to obtain $\lim_{(x,y) \to (0,0)} (x^2·y^3)/(x^4+y^6)$

There is no limit. Along the semicubical parabola $x=t^3$, $y=t^2$ the limit is equal to 1/2. S.G.

Note that if you replace $x^2$ by $x'$ and $y^3$ by $y'$, then the expression reduces to $\frac{x'y'}{x'^2 + y'^2}$.

If you now take $x' = y'$, then you get $1/2$. This shows that the limit does in fact not exist, as you found correctly that some approaches lead to $0$.

If you do not like the replacement: when you approach via $(t^3, t^2)$ you get $1/2$.