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A system can only be called semiclassical if there are parts of the system that are described classically and parts decsribed quantum-mechanically. In this paradigm, physical quantites are described in a power series of $\hbar$, with the zero order corresponding to classical physics and higher orders corresponding to quantum corrections.


Given the above, I do not see why the limit $\hbar \rightarrow 0$ called the semiclassical limit and not the classical limit. After all, in the limit that $\hbar \rightarrow 0$, the system becomes classical and there are no quantum-mechanical corrections.

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Semiclassical analysis is the name reserved for areas where some asymptotic approximation of a quantum mechanical object is employed. For example: in the Path Integral formalism you can take $\hbar$ to be very small, and this would mean that your integral will oscillate very rapidly and only contributions near the classical trajectory would be really relevant. This is called saddle point approximation. As a result you'll obtain a propagator that presents quantum characteristics (that it, it includes the possibility of state superposition) but depends on the classical system's solutions, which means it is deeply rooted in classical mechanics. This is the semiclassical propagator.

Remember that systems like the three-body problem are classically chaotic, which means we cannot solve the equations of motion without greatly simplifying them and applying numerical methods (in general). But the Helium atom, which is the quantum equivalent of a three-body problem, is not chaotic: the Schrödinger equation is linear, and therefore prohibits all chaos in quantum mechanical systems. Where did the chaos leak to? How can we study the process of a classical system being quantized and its chaos, lost?

That's is just one reason to study semiclassical analysis. The limit $h \to 0$ that you quote is, indeed, the classical limit... The point is that in semiclassical analysis you do not take $\hbar \to 0$, you take $\hbar \to \epsilon$.

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