Look, this is kinda' news to me but it seems to fall out right.. So if we can obtain the Taylor series:
and then use long division:
note the first term is -3/8 and I assume the remainder will have powers of x so that as x->0, the limit is -3/8 which is the limit given by Mathematica.
I'm not entirely sure about the validity of this method with the other one since has a branch-point at the origin.