Is it only valid to approximate using dimensionless parameters? In physics we usually have to approximate the behaviour of physical systems. It is often said that if the behaviour of a system depends on a certain parameter with units, then it is meaningless to approximate the behaviour of that system as that parameter becomes small or large. Why is this the case? I'd say that one is considering the limit as the parameter becomes small/big relative to the chosen unit, and that the limiting value should be independent of the choice of units assuming that the quantity we are approximating is analytic in a neighbourhood of $0$ or $\infty$.
 A: You're right that it's technically possible to talk about "the limit where $m \to 0$". But it isn't necessarily so useful.
What if $m$ is the mass of the proton? If we use kg, it's certainly pretty darn small relative to our chosen unit! But there's a big difference between a result that's valid in the limit $m_{\mathrm{proton}} / m_{\mathrm{earth}} \to 0$ and a result that's valid in the limit $m_{\mathrm{proton}} / m_{\mathrm{electron}} \to 0$...
Using dimensionless variables makes it completely clear what "small enough" means, when we say a result holds if the mass of the proton is "small enough".
Put slightly differently, if we have a function of some dimensionful quantity $x$ which we approximate for small $x$ like
$$f{\left(x\right)} \approx 1  +  \left(\frac{x}{x_0} \right)+ \dots.$$
Yes, it's true that the limit $x \to 0$, $f \to 1$ no matter what units we have, but if we want to know whether or not we can get away with saying $ f = 1$, we want to know how small $x/x_0$ is, not how small $x$ is in some arbitrary units.
