Evaluate $\lim\limits_{x\to\ 0}(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)})$ 
Evaluate  $$\lim\limits_{x\to\ 0}\left(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)}\right)$$

It is easy with L'Hospital's rule, but takes too much time to calculate derivatives. What are some shorter methods?

The series expansions are
$\sin x = x - x^3/6...,\,\,\tan x = x + x^3/3...,\,\, \arctan x=x - x^3/3...$
Equate numerators 
$$\frac{(x+x^3/3)-(x-x^3/6)}{(x - x^3/6)(x + x^3/3)(x^3/3)}$$
$$\frac{x^3/2}{(x - x^3/6)(x + x^3/3)(x^3/3)}$$
We only care about the $x^3$ in the numerator
$$\frac{x^3/2}{x^3}$$
$$1/2$$
