I am wondering about the possible best case approximation and worst case approximation of irrational numbers. I think the appropriate formulation is whether there are functions $\hat{b}(q)$ and $\check{b}(q)$ such that for any $\alpha\in [0,1)\setminus \mathbb{Q}$ we have for only finitely many $q's$

$$ \check{b}(q) \leq \Big\vert \alpha-\frac{p}{q} \Big\vert \quad \text{and} \quad \Big\vert \alpha-\frac{p}{q} \Big\vert \leq \hat{b}(q) \quad \text{for some}\;\; p\in \mathbb{Z}. $$

By the Piegonhole principle, we know that $\check{b}(q)\geq \frac{1}{2q}$. Also, by  Liouville's theorem we can see that $\hat{b}(q)=o(q^{-n})$ for all $n$.
Are there more explicit asymptotic bounds ? Can we say something like $\hat{b}(q)=o(e^{-q})$ ? Can we say that $\check{b}(q)=\Theta(\frac{1}{q})$?

I tried to ask this questions on Math Stackexchange, but I had no  response. 
I assume something like this is known and I have not found this in a short search online. I am unused to these notions so I apologize if this is a silly question.

I would appreciate any insights on this.