Limit of function as it approaches origin Find the limit of $f$ or prove that limit does not exist as $(x,y)$ approaches origin point.

$$f(x,y) = \frac{2x}{x^2 + x+ y^2}$$

So far, I have used several methods, such as finding correct $2$ paths for nonexistence of limit or applying polar coordinates, but I could not solve it properly. Any help would be appreciated!
$$f(x,x)=\frac{2x}{2x^2+x}\rightarrow2.$$
$$f(x^2,x)=\frac{2x^2}{x^4+2x^2+x^2}\rightarrow\frac{2}{3},$$ which says that the limit does not exist.
$$f(x,x)=\frac{2x}{2x^2+x}\rightarrow2.$$
$$f(x^2,x)=\frac{2x^2}{x^4+2x^2+x^2}\rightarrow\frac{2}{3},$$ which says that the limit does not exist.