How to find the limit of a $2$ variables function.

Question

I need to find the limit of $\dfrac{\sin xy} {x^2+3y^2}$ as $(x,y)$ approaches $(0,0)$.

I tried to look at what the limit is when I come in from different directions to find a 'contradiction' but I can't seem to do it.

Any help with this would be very much appreciated.

Answer 1

Hint: Try looking at the limit as $(x,y)\to(0,0)$ along the line $y=kx$.

Is the limit independent of $k$?