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 some spaces with this property.