Limit as $(x,y,z)\to (0,0,0)$ of $f(x,y,z) = \dfrac{xy+yz+xz}{\sqrt{x^2+y^2+z^2}}$ To find this limit, I converted to spherical coordinates and rewrote:
$$\lim_{r\to 0} \dfrac{r^2(\sin^2\theta \cos\phi \sin \phi + \sin\theta \cos \theta \sin \phi + \sin\theta \cos \theta \cos \phi)}{r} = 0$$
Is this method alright? Our teacher did using epsilon delta proof, so how can we use something similar to spherical coordinates if say we had four variable limit of kind:
$$\lim_{(w,x,y,z) \to (0,0,0,0)} \frac{xy+yz+xz+wx}{
\sqrt{x^2+y^2+z^2+w^2}}$$

Notice that $x,y$ and $z$ are less and equal than $\sqrt{x^2+y^2+z^2}$ so that
$$ |xy+yz+xz| \leq 3\left(\sqrt{x^2+y^2+z^2}\right)^2.$$
Hence, by a straightforward application of sandwich theorem, there holds that the aforementioned limit equals zero, since
$$ 0\leq \lim_{(x,y,z)\rightarrow (0,0,0)}\left|\dfrac{xy+yz+xz}{\sqrt{x^2+y^2+z^2}}-0 \right|\leq \lim_{(x,y,z)\rightarrow (0,0,0)}3\sqrt{x^2+y^2+z^2}=0.$$ 
