Prove the existance of the following limit: $\frac{xy+yz+zx}{x^2+y^2+z^2}$ Find the following limit: $$\lim_{(x,y,z) \rightarrow (0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^2}$$
I have completely no idea what to do with this. Use iterated limits and this will be enough? How to solve that?
 A: I think there is no limit here. For example take $x(t),y(t),z(t) = k_1 t,k_2t,k_3t$ and the expression will be different for different $k_1,k_2,k_3$.
A: The limit do not exists.
Take $z=0$ and $y=kx$, then the "limit" depends on $k$ and therefore do not exist.
A: Converting to spherical coordinate:
$$\lim_{(x,y,z) \rightarrow (0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^2}$$
$$ = \lim_{r \rightarrow 0} \frac{r^2\sin^2(\theta)\cos(\phi)\sin(\phi)+r^2\sin(\theta)\cos(\theta)\cos(\phi)+r^2\sin(\theta)\sin(\phi)\cos(\theta)}{r^2}$$
$$=\sin^2(\theta)\cos(\phi)\sin(\phi)+\sin(\theta)\cos(\theta)\cos(\phi)+\sin(\theta)\sin(\phi)\cos(\theta) $$
So I think since the the result depends to $\theta $ and $\phi$, the limit is not exist.
A: Just so that you have a really clear understanding of the specific values that your function takes, let $v= (x, y, z)^T$ and then your function is $$ \frac{1}{|v|^2} v^T \begin{pmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{pmatrix} v = \left(\frac{v}{|v|}\right)^T A \left(\frac{v}{|v|}\right)$$ where $A$ is the matrix in the previous expression.
Note that the function is constant on sets of the form $\{ t v : t\in \mathbb{R}^+, v\in \mathbb{R}^3\}$. Therefore, we may assume that $v$ is of unit length. Also, $A$ is a rotation by $2\pi/3$ about the vector $(1, 1, 1)^T$. When $v$ is parallel to $(1, 1, 1)^T$, we have the value $v\cdot v/|v|^2 = 1$, but when we are in the plane perpendicular to $(1, 1, 1)^T$, the value is $\cos(2\pi/3) = -1/2$. In fact, if we write our unit vector $$v = \frac{\cos(\theta)}{\sqrt{3}} (1, 1, 1)^T + \sin(\theta) w$$ where $w\cdot(1, 1, 1) = 0$ and $|w|=1$, then the value of your expression is $\cos^2(\theta) -\sin^2(\theta)/2 = 1- \frac{3}{2}\sin^2(\theta)$.
That's a lot of variation, and it happens in every sphere about the origin.
