Hey, hope this is in the right topic

\(\displaystyle f(x,y) = (sin(xy)-(xy))/sin(x^2+y^2)\)

Thats whats troubling me, I just really dont know where to start and I cant work it out from the solutions.

Thanks guys.(And gals)

Let's prove the limit is 0. We know that \(\displaystyle \sin(u)-u\sim-\frac{u^3}{6}\) when \(\displaystyle u\to0\). Since \(\displaystyle xy\to 0\) when \(\displaystyle (x,y)\to(0,0)\), we have \(\displaystyle \sin(xy)-xy\sim-\frac{(xy)^3}{6}\) when \(\displaystyle (x,y)\to(0,0)\). Similarly, \(\displaystyle \sin(x^2+y^2)\sim x^2+y^2\), so that \(\displaystyle f(x,y)\sim-\frac{(xy)^3}{6(x^2+y^2)}\) when \(\displaystyle (x,y)\to(0,0)\).

Now we can say for instance \(\displaystyle \frac{|xy|^3}{x^2+y^2}\leq |xy^3|\frac{x^2}{x^2+y^2}\leq |xy^3|\) (since the ratio is less than 1) hence \(\displaystyle \frac{(xy)^3}{6(x^2+y^2)}\to0\)when \(\displaystyle (x,y)\to(0,0)\). This concludes.