I am trying to learn about asymptotic expansions. I am using the book Singular Perturbation Theory by R.S. Johnson. I have now stumbled upon an expansion that I do not understand (eq. 1.68 in the book). Here, it is stated that we may expand (with $x=O(1)$)
$$ \frac{1}{1+e^{-x/\epsilon}}\left(1-\frac{x+\epsilon}{1+e^{-x/\epsilon}}\right)\sim (1-e^{-x/\epsilon})(1-x-\epsilon)+(x+\epsilon)e^{-x/\epsilon} $$
where we retain terms $O(1)$, $O(\epsilon)$, $O(e^{-x/\epsilon})$ and $O(\epsilon e^{-x/\epsilon})$.
How does one go about reproducing this expansion? I have tried expanding it in a Taylor series around $\epsilon=0$, but it does not lead to the same result. Any guidance would be greatly appreciated.