TITLE: Find limits of a function with several variables
QUESTION [5 upvotes]: Does this $$\lim_{x,y,z\to(0,0,0)}\frac{xy+2xz+yz}{{x^2+y^2+z^2}}$$ have a limit?
My answer for this is
Let f(x,y,z)=$$\frac{xy+2xz+yz}{{x^2+y^2+z^2}}$$
then,
$$\lim_{x\to0}{f(x,0,0)}=\lim_{x\to0}\frac{0}{x^2}=0$$
$$\lim_{x\to0}{f(x,x,0)}=\lim_{x\to0}\frac{x^2}{2x^2}=\frac{1}{2}$$
Since this two limit are not the same,$$\lim_{x,y,z\to(0,0,0)}\frac{xy+2xz+yz}{{x^2+y^2+z^2}}$$ does not exist.
I'm not sure if this justification is enough or correct.

REPLY [2 votes]: Recalling spherical coordinates
$$ x = \rho \cos(\theta)\sin(\phi),\, y=\rho \sin(\theta)\sin(\phi),\, z =\rho \cos(\phi), \quad 0\leq \theta \leq 2\pi,\, 0\leq \phi \leq \pi. $$
we have
$$ \frac{xy+2xz+yz}{{x^2+y^2+z^2}}= \cos(\theta)\sin(\theta)\sin^2(\phi)+2\cos(\theta)\sin(\phi)\cos(\phi)+ \sin(\theta)\sin(\phi)\cos(\phi). $$ 
Now the above expression achieves an infinite number of values depending on $\theta$ and $\phi$ which implies the limit does not exist.