Continuity of $f(x,y) = \dfrac{\sin(x^2+y^2)}{x^2+y^2}$ at $(x,y) = (0,0)$ Continuity of $f(x,y) = \dfrac{\sin(x^2+y^2)}{x^2+y^2}$ at $(x,y) = (0,0)$
I need to specify the value of $f(x,y)$ which makes the given function continuous.
I had tried to give a arbitral relationship between $y$ and $x$, but as $(x, y)$ gets closer to $(0, 0)$ the arbitral movement cannot always be represented as a functional representation between $y$ and $x$.
How could I solve this problem rigorously?

Recalling that $|x|\,|\cos(x)|\le |\sin(x)|\le |x|$ for $0\le |x|\le \pi/2$, we assert that for $x^2+y^2\le \pi/2$
$$|\cos(x^2+y^2)|\le \left|\frac{\sin(x^2+y^2)}{x^2+y^2}\right|\le 1$$
whence application of the squeeze theorem reveals
$$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=1$$
