Show that $\lim_{(x,y) \to (0,0)} {\frac{x\sin y – y\sin x}{x^2 +y^2}}$ does not exist
I did use Wolfram Alpha and it says this limit does not exist.
I'm trying to prove this with sequential definition of multivariable function. So basically, I have to find two sequences $(u_k)$ and $(v_k)$ such that they approach $(0,0)$ but the two sequences $f(u_k)$ and $f(v_k)$ approach two different limits. I tried various things but nothing works out. Could you give me some hint about this problem? Thank you in advance!
Best Answer
For $x,y\ne 0$, we have:
$$\frac{x\sin y-y\sin x}{x^2+y^2}=\left(\frac{\sin y}{y}-\frac{\sin x}{x}\right)\frac{xy}{x^2+y^2}$$
so when $(x,y)\to (0,0)$, the first factor $\frac{\sin y}{y}-\frac{\sin x}{x}$ converges to $1-1=0$ and the second factor is bounded:
$$|xy|\le\frac{1}{2}(x^2+y^2)\implies\left|\frac{xy}{x^2+y^2}\right|\le \frac{1}{2}$$
so the whole function converges to zero. The same happens on the lines $x=0$ or $y=0$ (as the function is zero there).
So, it seems to me that $\lim_{(x,y)\to(0,0)}\frac{x\sin y-y\sin x}{x^2+y^2}$ exists after all, and it is $0$.