Let $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$. STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$. and STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$. (A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True
Let $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$.
STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$.\\
and\\
STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$.\\
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1\\
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1\\
(C) STATEMENT-1 is True, STATEMENT-2 is False\\
(D) STATEMENT-1 is False, STATEMENT-2 is True