We have
$$ L = \lim_{(x,y)\to(0,0)} \frac{\sin(x^2+y^2)}{6x^2+6y^2} $$
Let $x = r\cos\theta$ and $y = r\sin\theta$. Then $x^2+y^2 = r^2$. As $(x,y)\to(0,0)$, $r\to 0$.
Then
$$ L = \lim_{r\to 0} \frac{\sin(r^2)}{6r^2} $$
We know that $\lim_{u\to 0} \frac{\sin u}{u}
Show more…