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:
$$\lim_{(w,x,y,z) \to (0,0,0,0)} \frac{xy+yz+xz+wx}{
\sqrt{x^2+y^2+z^2+w^2}}$$

Using the full spherical coordinates is overkill here. Let $r=\sqrt{x^2+y^2+z^2}$.
Then $|x|\le r$, $|y|\le r$, $|z|\le r$. So
$$|xy+xz+yz|\le|xy|+|xz|+|yz|\le 3r^2$$ and so
$$\left|\frac{xy+xz+yz}{\sqrt{x^2+y^2+z^2}}\right|\le 3r.$$
As $\lim_{(x,y,z)\to(0,0,0)}r= 0$ then
$$\lim_{(x,y,z)\to(0,0,0)}\left|\frac{xy+xz+yz}{\sqrt{x^2+y^2+z^2}}\right|=0$$
also.
This method works for your four-variable problem too, avoiding
the minutiae of four-dimensional spherical coordinates.
