All depends on how that cube root is done. To get surd behavior one requires CubeRoot (or Surd) in Mathematica.
In[154]:= Limit[(x + 5*Sqrt[4 + x^2])/CubeRoot[9 + 8*x^3],
x -> -Infinity]
(* Out[154]= -2 *)
In[155]:= Limit[(x + 5*Sqrt[4 + x^2])/(9 + 8*x^3)^(1/3),
x -> -Infinity]
(* Out[155]= 1 - I Sqrt[3] *)
As for Wolfram|Alpha, I do not know offhand what heuristics it uses to determine which was intended. But one can force the surd behavior by using cbrt().
lim (x to -infinity ) ((x+5sqr(4+x.b2))/cbrt(9+8x.b3))
