I'm trying to understand permutation theory and in particular the $$z=\frac{x}{\epsilon}$$ substitution to get an inner solution.
Here is my toy example: $$f(x)=\sqrt{x} - x^{1/\epsilon}$$ and I'm interested in the interval $[0,1]$.
Does it count as two time scales and a boundary layer at $x=1$?
So we balance the original and, as I presume because in $[0,1)$ we have $$x^{1 / \epsilon} \in o \left(\sqrt x \right), \text{ as } \epsilon \to 0,$$ we drop it and get $$f_\text{outer} = \sqrt x.$$
Now we substitute $$z=\frac{x-1}{\epsilon}$$ ($x-1$ because the layer is at $1$) and get $$f(z)=\sqrt{\epsilon z +1} - (\epsilon z +1)^{1 / \epsilon}.$$
The two terms don't seem to be of different order as $\epsilon \to 0$. How can we get $f_\text{inner}$?