Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z\_n\}$ such that $\lim\_{n\to +\infty } d(x,z\_n)=+\infty$ and $\lim\_{n \to +\infty } \frac{d(x,z\_n)}{d(y,z\_n)}=1 $. 


Note: I should say, I know this property is true for some spaces like quasi normed spaces, and I want to know more about other spaces with this property.



A property on some unbounded metric spaces
Unboundedness guarantees that there is one sequence $(z\_n)\_n$ such that $d(x,z\_n)\to\infty$ for all $x$.
That sequence also satisfies the second requirement via the triangle inequality:
$$
\frac{d(x,z\_n)}{d(y,z\_n)}\le \frac{d(x,y)}{d(y,z\_n)}+1
$$
and symmetrically
$$
\frac{d(y,z\_n)}{d(x,z\_n)}\le \frac{d(y,x)}{d(x,z\_n)}+1
$$
which shows that the limit of the quotient is equal to $1$


