II-A- The function $f$ defined on $\mathbb { R } ^ { * }$ by $f ( x ) = e ^ { \frac { 1 } { x } }$ has derivative $f ^ { \prime } ( x ) = e ^ { \frac { 1 } { x } }$. II-B- The function $F$ defined on $[ 0 ; + \infty [$ by $F ( x ) = x \sqrt { x }$ is an antiderivative of the function $f$ defined by $f ( x ) = \frac { 3 } { 2 } \sqrt { x }$. II-C- The function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = ( \ln ( 3 x ) ) ^ { 2 }$ has derivative $f ^ { \prime } ( x ) = \frac { 2 } { 3 x } \ln ( 3 x )$. II-D- $\quad \lim _ { x \rightarrow 0 } ( x \ln ( x ) - x ) = - \infty$. II-E- $\quad \lim _ { x \rightarrow + \infty } \left( x e ^ { x } - \ln ( x ) \right) = 0$. For each statement, indicate whether it is TRUE or FALSE.
\section*{Exercise II}
II-A- The function $f$ defined on $\mathbb { R } ^ { * }$ by $f ( x ) = e ^ { \frac { 1 } { x } }$ has derivative $f ^ { \prime } ( x ) = e ^ { \frac { 1 } { x } }$.\\
II-B- The function $F$ defined on $[ 0 ; + \infty [$ by $F ( x ) = x \sqrt { x }$ is an antiderivative of the function $f$ defined by $f ( x ) = \frac { 3 } { 2 } \sqrt { x }$.\\
II-C- The function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = ( \ln ( 3 x ) ) ^ { 2 }$ has derivative $f ^ { \prime } ( x ) = \frac { 2 } { 3 x } \ln ( 3 x )$.\\
II-D- $\quad \lim _ { x \rightarrow 0 } ( x \ln ( x ) - x ) = - \infty$.\\
II-E- $\quad \lim _ { x \rightarrow + \infty } \left( x e ^ { x } - \ln ( x ) \right) = 0$.
For each statement, indicate whether it is TRUE or FALSE.