Motivation of the singular perturbation solution formulation for local volatility model

I am puzzled by the motivation of the particular choice of the (singular) perturbation method used in Equivalent Black Volatilities. Equation (A.6a) sets $$\epsilon:= A(K)\ll 1.$$ What is the motivation for this setting? I find it surprising that $A(K)$ is to be infinitesimal. However, at later expansion in Equation (A.9a), $A(K)$ seems to be treated independently from $\epsilon$ which is $A(K)$ itself. Moreover, Equation (A.9b) seems to assume $A'(K)$ and $A''(K)$ to be infinitesimal as well, if $\nu_1$ and $\nu_2$ are to be finite. This setting seems to be rather contrived.

What is going on?

This local volatility analysis is referenced in the paper Managing Smile Risk on the SABR model.

Answer

In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let $x:=\frac{f-K}\epsilon$. The rest is the usual singular perturbation operation.