I'm trying to use L'Hopital's rule to calculate:
$$\lim_{x \to 0^+} \dfrac{x - \sin x}{(x \sin x)^{(3/2)}}$$
Taking a couple of derivatives of the denominator gets quite nasty, so I'd like to find a simpler way to do it.
I would like to make a change of variable, say, $t = \sqrt{x \sin x}$, to get a $t^3$ in the denominator. Unfortunately, that leaves me with problems in the numerator. Maybe there is some other manipulation or some trig identity that simplifies things that I am missing? This shouldn't be a difficult problem, but I can't seem to find a slick way to do it.
The answer is given as $\frac{1}{6}$. Thanks for your help.