Let f(x)=sinπ6sinπ2sinx for all x∈R and g(x)=π2sinx for all x∈R.  Let (f∘g)(x) denote f(g(x)) and (g∘f)(x) denote g(f(x))then which of the following is true statement.

We have1) f(x)=sinπ6sinπ2sinx-1≤sinπ2sinx≤1⇒-π6≤π6sinπ2sinx≤π6⇒-12≤sinπ6sinπ2(sinx)≤12 ∴range of fx is -12,122) f(g(x))=fπ2sinx=sinπ6sinπ2sinπ2sinx ∴the range of fogx is -12,123) limx→0fxgx=limx→0sinπ6sinπ2sinxπ2sinx                 =limx→0cosπ6sinπ2sinxπ6cosπ2sinxπ2cosxπ2cosx                 =π64) Assume ∃ x∈R such that gofx=1⇒π2sinfx=1 ⇒sinsinπ6sinπ2sinx=2π>12which is not possible ∴options (1), (2) and (3) are correct