JEE-Advanced 2015 (17/40)

For real $x$ , define $f(x)=\sin \left( \frac{\pi}{6} \sin \left( \frac{\pi}{2} \sin x \right) \right)$ and $g(x)=\frac{\pi}{2} \sin x$ . Let $(fog)(x)$ denotes $f(g(x))$ and $(gof)(x)$ denotes $g(f(x))$ . Then which of the following is/are true?

$\begin{array} { r l } (1) & \text{ The range of }f \text{ is } [-\tfrac12, \tfrac12] \\ (2) & \text{ The range of } f\circ g \text{ is } [-\frac12, \frac12 ] \\ (3) & \lim \limits_{x\to0} \frac{f(x)}{g(x)} = \frac\pi6 \\ (4) & \text{ There is an }x\in\mathbb R \text{ such that } f \circ g (x) = 1 \end{array}$

Submit your answer as the increasing order of the serial numbers of all the correct options. For example, if your answer is $(1),(2)$ , then submit 12 as the correct answer, if your answer is $(2),(3),(4)$ , then submit 234 as the correct answer.