# Limit and continuity of multivariable function

$g(x,y) = \left\{ \begin{array}{cl}\dfrac{\sin(2x^2 +2y^2)}{x^2 + y^2} & \mbox{if$(x,y)\neq(0,0)$}\\ 0 &\mbox{if$(x,y) = (0,0)$}\end{array}\right.$

For what values of a, if any, is $g(x,y)$ continuous at $(0,0)?$

How can we approach this problem? I'm thinking of proving this limit exist but it's quite complicated since that has not been covered in my course. I also tried to prove this limit does not exist but the result was $2$.

• hint : $\frac {sin(u)}{u}\to 1$ near $0$. – zwim Sep 22 '17 at 1:52
• And where is the $a$? – user480281 Sep 22 '17 at 1:52
• I used L'Hopital's Rule to prove it exists and limit goes to 2. I dont know if it's correct – Matthew Sep 22 '17 at 1:54
• Indeed, it is correct. – zwim Sep 22 '17 at 1:55

In polar coordinates $$f(r,\theta) = \frac{\sin(2r^2)}{r^2}.$$ Any path that approaches the origin has $r\to 0.$