Evaluate the limit containing $\arctan{x}$ and $\arcsin{x}$ Evaluate:
$$\lim_{x\to{0}}\bigg(\frac{2}{x^3}.(\arcsin{x}-\arctan{x})\bigg)^{2/x^2}$$
I can just expand $\arcsin{x}$ and $\arctan{x}$ using their taylor expansions, but is there any other method?

From https://math.stackexchange.com/a/198337/62967 we have
$$
\arcsin x = x + \frac{x^3}{6} + \frac{3x^5}{40} + o(x^5)
$$
and we know that
$$
\arctan x = x -\frac{x^3}{3} + \frac{x^5}{5} + o(x^5)
$$
Therefore
$$
\frac{2(\arcsin x-\arctan x)}{x^3}=1-\frac{1}{4}x^2+o(x^2)
$$
The logarithm of your function can therefore be written as
$$
\frac{2}{x^2}\log\left(1-\frac{1}{4}x^2+o(x^2)\right)
=
\frac{2}{x^2}\left(-\frac{1}{4}x^2+o(x^2)\right)=-\frac{1}{2}+o(1)
$$
Can you finish?
I see no other practical method.
