ω=((((a^2)+a b+(b^2)))^(1/2));
λ=FullSimplify[(((Sinh[a+b])/(a+b))(ω/(Sinh[ω]))-1)/(a b)]

λ/.b->0

Assuming[a>0,Limit[λ,b->0,Direction->"FromAbove"]]

ser=Collect[
Normal@Series[λ/.{a->a ℏ, b->b ℏ}, {ℏ,0,14}],
ℏ,Simplify]

ser/.b->0

Series[Log[(x/((E^x)-(E^(-x))))], {x,0,16}]

638512875/2730

2605132530000/325641566250