$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 $2$.

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


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