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

Hint: Consider the limit along the paths $t \mapsto (t,t,t)$ and $t\mapsto (t,0,0)$.

It is an overkill for sure, but since the quadratic form associated with $xy+xz+yz$ has zero trace, there is a positive eigenvalue and a negative eigenvalue too, so the ratio is positive along a direction and negative along a different direction, and the limit cannot exist.

Once consider x=t and y=t and z=t and then x=t and y=t and z=2t. In first case limit is 1 and in second case 5/6.So limit does not exist.