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.
Why not skeeter? They're both analytic at zero so it's easy to compute the first few terms of the Taylor series for each just be taking a few derivatives at x=0. I guess all this is ok. Never solved a limit this way before. Maybe some issues of uniform convergence is required for it to be valid though.