Answer
Limit does not exist
Work Step by Step
$ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \dfrac{2x}{x^2+x+y^2}$
Use polar-coordinates:
$x= r \cos \theta , y = r \sin \theta\\r^2=x^2+y^2$
or, $\lim\limits_{(x,y) \to (0,0) } \dfrac{2x}{x^2+x+y^2}=\lim\limits_{r \to 0} \dfrac{ 2 r \cos \theta}{r^2+r\cos \theta} =\lim\limits_{r \to 0} \dfrac{ 2 \cos \theta}{r+\cos \theta}$
The limit does not exist for $\cos \theta=0$
Thus, we can see that
$ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \dfrac{2x}{x^2+x+y^2}=DNE$
