Answer
$\dfrac{\pi}{2}$
Work Step by Step
The polar-coordinates are defined as: $x= r \cos \theta , y = r \sin \theta$ and $r^2=x^2+y^2$
$ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } arctan (\dfrac{|x|+|y|}{x^2+x+y^2}$
or, $=\lim\limits_{r \to 0} arctan (\dfrac{ |r \cos \theta| +|r \sin \theta|}{r^2})$
or, $=\lim\limits_{r \to 0} arctan (\dfrac{ |\cos \theta| +|\sin \theta|}{r})$
or, $=arctan (\infty)$
or, $ =\dfrac{\pi}{2}$
