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.