I hope this answers your question. Let me preface this by saying that there are (probably) no satisfying answers for these questions, and that I'm not an expert. There is a classical limit <https://en.wikipedia.org/wiki/Classical_limit> that will recover classical equations from quantum equations from the limit ℏ -> 0. Such a thing is a heuristic, which means that we just know some equations/models where it works, but have not discovered a general truth. There are also situations where you may take c -> +∞ for example, and that would be called the non-relativistic limit. Why do we take these limits? Because when we did, the answer was not complete nonsense. We don't know what to make of them, i.e. we don't have complete theories. Also, what these limits mean is not a simple matter of calculus, they are not point-wise limits.

In one such instance I've been studying for years, the WKB approximation, I've realized two things: 1) the approximations are not well understood and 2) the mathematics are quite complicated, but these points notwithstanding the equations are used in experiments. You can read the few-page introduction in "Lectures on the Geometry of Quantization" by Bates & Weinstein <https://math.berkeley.edu/~alanw/GofQ.pdf> to see some of this, in particular the subsection "Quantization and the classical limit". I'll just quote the relevant paragraph:

> Although there remain some unsettled issues connected with the question, “How can ℏ become small?” the answer is essentially the following. For any particular mechanical system, there are usually characteristic distances, masses, velocities, . . . from which a unit of action appropriate to the system can be derived, and the classical limit is applicable when ℏ divided by this unit is much less than 1.

But remember, this is just one approach to the subject. Another heuristic is this: h has dimensions energy x time, which means it converts frequency into energy, e.g. E = hf. In the Fourier transform, the character is exp(2πihx·ξ), where ξ is the frequency. The effect of h -> 0 would be to dampen high-energy waves. Irregularity comes from high frequencies (think of it like this: a sum of sines of large periods would not have many kinks.) When you "iron out" the irregularity of the quantum solution, you end up with a classical one.

Again disclaimer: not a physicist, nor an expert.