question_answer

Let \[f\,\,(x)=\sin \,\,\left( \frac{\pi }{6}\,\,\sin \,\,(\frac{\pi }{2}\sin \,\,x) \right)\]  for all \[x\in R\] and \[g(x)=\frac{\pi }{2}\sin x\] for all \[x\in R.\] Let (fog) (x) denote \[f\,\,(g\,\,(x))\] and \[(gof)\,\,(x)\] denote \[g\,\,(f\,\,(x)).\]Then which of the following is false?  

A)  \[\underset{x\to 0}{\mathop{\lim }}\,\,\,\frac{f\,\,(x)}{g\,\,(x)}=\frac{\pi }{6}\]

B)  Range off is \[\left[ -\frac{1}{2},\,\,\frac{1}{2} \right]\]

C)  Range of fog is \[\left[ -\frac{1}{2},\,\,\frac{1}{2} \right]\]

D)  There is an \[x\in R\] such that \[\left( gof \right)\left( x \right)=1\]

E)  None of these