lim (1+tanx)^(1/2) - (1+sinx)^(1/2)
x->0 _____________________________
x^{3 how do you do this? the conjugate is no help!}
Unfortunately, you have apparently already decided that "the conjugate is no help" and that is certainly the method I would suggest! It also helps to know the basic limit, $\displaystyle \lim_{x\to 0} sin(x)/x= 1$.
Or, just calculating that quantity for, say x= 0.00001 should give you a good idea.