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)
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.