Is the correspondence principle intentionally a physical statement regarding perturbation theory? As a new physics reader I accidentally stumbled across perturbation theory as an idea to relate the modeling tactics I see repeatedly employed in

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*Statistical mechanics to approximate molecular motion in a solid disregarding high-order terms of van der Waals interaction


*$\tan, \sin\theta \approx \theta$ in the scope of optics, and pendulums to reduce things to elementary functions


*Kinetic energy in light of special relativity.
However it has now been brought to my attention that these may all also be considered instances of correspondence principles.
... to draw a formal analogy between physics and mathematics is this the same as
superposition : addition :: correspondence : approximation?
Also where can I read more concentrated goodness about the broad spectrum of theoretical antics like this?
 A: I would say there's an indirect link, but the correspondence principle is more "fundamental" whereas perturbation theory is "just" an (important) approximation scheme.
The underlying physical idea is that physics should be continuous (actually, analytic) if we vary a continuous parameter. This might seem like a strong assumption, but it fits with the general intuition you might have that you shouldn't be able to say whether a parameter is exactly zero, or just so close to zero that our experimental uncertainty is too large to tell the difference with zero. (However, there are cases where physics can change discontinuously with a parameter, for example the number of degrees of freedom of a photon changes from 3 to 2 when the mass of the photon is small vs exactly zero).
Strictly speaking we don't ever get to vary a parameter like $\hbar$. But, usually in any specific example, we can define a dimensionless parameter, proportional to $\hbar$, which goes to zero in the classical limit. For example, the difference between the quantized electric field and the classical electric field goes to zero when the number of photons is large; you can think of this as being due to the fact that the coefficient of variation $\sim 1/\sqrt{N}$ (where $N$ is the number of photons) is tending toward zero.
The correspondence principle then concerns limits in parameter space.
Perturbation theory involves doing a Taylor expansion in parameter space, and keeping the first few terms. The limit should correspond to the leading order behavior in the Taylor series, if the analyticity assumption is correct.
