Consider $$f(x,y )=\frac{x^2y^2}{2x^2+y^2}$$
Using polar coodinates we get $$f(r,\varphi)=\frac{r^4\cos(\varphi)^2\sin(\varphi)^2}{2r^2\cos(\varphi)^2 + r^2\sin(\varphi)^2}$$
Now: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{r\to 0} f(r,\varphi) = 0$$ so $f(x,y)$ is indeed continuous since it isn't dependent on $\varphi$.
Now let's consider $\displaystyle g(x)=\frac{x^2y^2}{2(x+1)^2+(y+1)^2}$
The point which makes us cringe is $(-1,-1)$. Now, if I'd want to use the polar coordinates method, how would I do it? Would I substitute $u=x+1$, $v=y+1$ to get $r\to 0$ again? If not, what would I do?