Given $f ( x ) = \frac { \ln ( x ) } { x }$, where ln denotes the natural logarithm, defined for $\mathrm { x } > 0$, find: a) ( 0.5 points) Calculate, if it exists, a horizontal asymptote of the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$. b) (1 point) Find a point on the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ where the tangent line to the curve is horizontal and analyze whether this point is a relative extremum. c) (1 point) Calculate the area of the bounded region limited by the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ and the lines $\boldsymbol { y } = \mathbf { 0 }$ and $\boldsymbol { x } = \boldsymbol { e }$.
Given $f ( x ) = \frac { \ln ( x ) } { x }$, where ln denotes the natural logarithm, defined for $\mathrm { x } > 0$, find:
a) ( 0.5 points) Calculate, if it exists, a horizontal asymptote of the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$.
b) (1 point) Find a point on the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ where the tangent line to the curve is horizontal and analyze whether this point is a relative extremum.
c) (1 point) Calculate the area of the bounded region limited by the curve $\boldsymbol { y } = \boldsymbol { f } ( \boldsymbol { x } )$ and the lines $\boldsymbol { y } = \mathbf { 0 }$ and $\boldsymbol { x } = \boldsymbol { e }$.