In the question above, I understand that to compute the outer layer you take x = O(1). Thus this means in the asymptotic expansion the first term disappears since it is so small.
However, there is then no term of order epsilon since you've removed that the only term that contained an epsilon. Where have I gone wrong here?
Additionally, when computing the inner expansion, how do you deal with the coth term, is it via exp expansion then taylor expansion?
Any help would be greatly appreciated. Thank you!
a) You've done nothing wrong in calculating the 'outer layer': the $\epsilon$-dependent term is exponentially small in $\epsilon$, i.e. smaller than any power of $\epsilon$, or 'beyond all orders', however you want to call it. It just identically vanishes in this approximation region.
b) For the 'inner layer', i.e. taking $x = \epsilon \xi$, I think it is indeed the easiest to first expand the exp-functions in the numerator and denominator of the hyperbolic cotangent, and then use a Taylor expansion.