Interesting. Applying LHopital's rule twice gives -1/2:
expr = (Log[1 - x] - Sin[x])/(1 - Cos[x]^2);
d2num = D[Numerator[expr], {x, 2}]
(* -(1/(1-x)^2)+Sin[x] *)
d2denom = D[Denominator[expr], {x, 2}]
(* 2 Cos[x]^2-2 Sin[x]^2 *)
d2num/d2denom /. x -> 0
(* -(1/2) *)
But this is not correct, so LHopital's rule must arrive at an indeterminate form. But why is this indeterminate?
