Answer
$\lim\limits_{x\to 0}$ is undefined.
Work Step by Step
A graph of the function $f(x) = \frac{sin(tan(x))-tan(sin(x))}{arcsin(arctan(x)) -arctan(arcsin(x))}$ is shown below.
Notice that as x approaches 0, the function does not approach any one value, so therefore, the limit does not exist.
