Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two functions. Consider the following two statements: $\mathbf { P ( 1 ) }$: If $\lim _ { x \rightarrow 0 } f ( x )$ exists and $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$ exists, then $\lim _ { x \rightarrow 0 } g ( x )$ must exist. $\mathbf { P ( 2 ) }$: If $f , g$ are differentiable with $f ( x ) < g ( x )$ for every real number $x$, then $f ^ { \prime } ( x ) < g ^ { \prime } ( x )$ for all $x$. Then, which one of the following is a correct statement? (A) Both $\mathrm { P } ( 1 )$ and $\mathrm { P } ( 2 )$ are true. (B) Both $P ( 1 )$ and $P ( 2 )$ are false. (C) $\mathrm { P } ( 1 )$ is true and $\mathrm { P } ( 2 )$ is false. (D) $\mathrm { P } ( 1 )$ is false and $\mathrm { P } ( 2 )$ is true.
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two functions. Consider the following two statements:\\
$\mathbf { P ( 1 ) }$: If $\lim _ { x \rightarrow 0 } f ( x )$ exists and $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$ exists, then $\lim _ { x \rightarrow 0 } g ( x )$ must exist.\\
$\mathbf { P ( 2 ) }$: If $f , g$ are differentiable with $f ( x ) < g ( x )$ for every real number $x$, then $f ^ { \prime } ( x ) < g ^ { \prime } ( x )$ for all $x$.\\
Then, which one of the following is a correct statement?\\
(A) Both $\mathrm { P } ( 1 )$ and $\mathrm { P } ( 2 )$ are true.\\
(B) Both $P ( 1 )$ and $P ( 2 )$ are false.\\
(C) $\mathrm { P } ( 1 )$ is true and $\mathrm { P } ( 2 )$ is false.\\
(D) $\mathrm { P } ( 1 )$ is false and $\mathrm { P } ( 2 )$ is true.