what is limit of $\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(2x+2y)-2x-2y}{\root\of {{x^2+y^2}}}$

Question

Limit of $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(2x+2y)-2x-2y}{\root\of {{x^2+y^2}}}$$

How can evaluate this limit?
I tried using polar coordinates like $x=r\cos\theta$ and $y=r\sin\theta$. put the thing in the sin doesn't turn into something neat.
Also i tried evaluating along different paths. But I don't know how that helps.

Edit: Also could you prove that the limit you found is the actual limit using the $\epsilon - \delta$ method.

Thanks!

Answer 1

HINT

We have that

$$\frac{\sin(2x+2y)-2x-2y}{\root\of {{x^2+y^2}}}=\frac{2x+2y}{\root\of {{x^2+y^2}}}\cdot \left(\frac{\sin(2x+2y)}{2x+2y}-1\right)$$