When $z \to 0$, can we approximate the hyperbolic distance between $z$ and $0$ by the usual Euclidean distance?

Question

Let $z \in \mathbb{C}$. Let $\rho$ represent the hyperbolic metric on the open unit disk and $|z|$ be the usual length of $z$, when $z \to 0$, can we say that $\rho(z , 0) \to |z|$?

I just want to know whether this is true. Some intuitive explanation would be greatly appreciated.