Why are we allowed to let $\hbar \ \rightarrow \ 0$ in the semi-classical regime? I am currently studying the WKB approximation, and certain parts of the argument (mostly when dealing with turning points and patching wavefunctions) rely on the fact that the WKB approximation is a semi-classical approximation, and in the semi-classical regime, $\hbar \ \rightarrow \ 0$. 
I understand how certain aspect of classical mechanics can be recovered as Planck's constant gets smaller, but my question is: why are we allowed to do this? After all, we are using the WKB approximation in the context of regular quantum mechanics, where $\hbar$ is just a fixed number. It doesn't really make sense to me why we are able to make this assumption.
 A: We never "let $\hbar \to 0$". As you said, that doesn't make sense because $\hbar$ is dimensionful, and also fixed in our universe.
What we mean by $\hbar \to 0$ is that we're considering only physical situations where the action $S$ is large, and thus taking the limit $\hbar/S \to 0$. That's what the semiclassical regime means.
It's like how the "nonrelativistic regime" means considering only objects with speeds $v$ small compared to the speed of light, $v/c \to 0$. Sometimes people sloppily write that as $c \to \infty$. 
A: What is happening is that the solution to the Schroedinger Equation is being Taylor series expanded in $\hbar$, and the semiclassical approximation is good when the first and higher order terms are relatively small, and so you can drop them (which is the same as setting $\hbar$ to zero) without affecting the form of the solution much. We are "allowed" to do this only when the higher order coefficients are near zero, and when they are that is the reason.
The physical reason is that when you get to large enough scales, $\hbar$ may as well be zero because of how little difference it makes. It's not exact, but this is an approximation so you already aren't exact. You're just making another assumption which restricts the regime of validity, and you just need to keep that in mind.
