Link between Quantum and Classical Mechanics [duplicate]

In classical mechanics we have momentum as generator of translation by following definition:

$$f(x+\delta x)=f(x)+[f(x),p]\delta x+....$$

I was wondering whether using this relation and commutation relation between $\hat{x}$ and $\hat{p}$ can we come to a quantum mechanic relation of momentum as generators:

$$\left(1-\frac{i}{\hbar}\hat{p}dx\right)|x\rangle=|x+dx\rangle$$

I am preferring to use commutation relationships as they are the bridge between quantum mechanics and classical mechanics.