Find the limit $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$ Find the limit if it exists
$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$$

Let's see the limit in the plane $z=0$. So,
\begin{align*}
     \lim_{(x, y, z)\rightarrow (0,0,0)} \frac{xy + yz + xz}{x^2 + y^2 + z^2} &= \lim_{(x, y) \rightarrow (0,0)} \frac{xy} {x^2 + y^2}
\end{align*}
Now analyzing by paths, we have to take the path $y=mx$, with $m>0$.
\begin{align*}
\require{cancel}
     \lim_{(x, y)\rightarrow (0,0)} \frac{xy}{x^2 + y^2} &= \lim_{x\rightarrow 0} \frac{mx^2}{x^2 + m^2x^2}\\
     & = \lim_{x\rightarrow 0} \frac{mx^2}{x^2(1 + m^2)}\\
     & = \lim_{x\rightarrow 0} \frac{m\cancel{x^2}}{\cancel{x^2}(1 + m^2)}\\
     & = \frac{m}{1 + m^2}.
\end{align*}
The above tells us that the limit is not unique, because it varies according to the value that $m$ takes. Therefore the limit $\displaystyle \lim_{(x, y, z) \rightarrow (0,0,0)}\frac{xy + yz + xz}{x^2 + y^2 + z^2}$ does not exist.
