A Double Limit Question Maybe it's easy, but: Is it true that
$$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$
If it is, could you help me prove it?
Thanks

Another approach, perhaps simpler: use polar coordinates
$$x=r\cos\theta\;,\;\;y=r\sin\theta\;,\;\;\text{and observe that}\;\;(x,y)\to 0\implies r\to 0\;,\;\;\text{so}$$
$$\frac x{\sqrt{x^2+y^2}}=\frac{r\cos\theta}{r}=\cos\theta\xrightarrow[r\to 0]{}\cos\theta$$
and the limit depends on $\;\theta\;$ and thus it cannot exist .
