On the coordinate plane, for a natural number $n$, let $\mathrm { A } _ { n }$ be the point where the two lines $y = \frac { 1 } { n } x$ and $x = n$ meet, and let $\mathrm { B } _ { n }$ be the point where the line $x = n$ and the $x$-axis meet. Let $\mathrm { C } _ { n }$ be the center of the circle inscribed in triangle $\mathrm { A } _ { n } \mathrm { OB } _ { n }$, and let $S _ { n }$ be the area of triangle $\mathrm { A } _ { n } \mathrm { OC } _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } } { n }$? [4 points] (1) $\frac { 1 } { 12 }$ (2) $\frac { 1 } { 6 }$ (3) $\frac { 1 } { 4 }$ (4) $\frac { 1 } { 3 }$ (5) $\frac { 5 } { 12 }$
On the coordinate plane, for a natural number $n$, let $\mathrm { A } _ { n }$ be the point where the two lines $y = \frac { 1 } { n } x$ and $x = n$ meet, and let $\mathrm { B } _ { n }$ be the point where the line $x = n$ and the $x$-axis meet. Let $\mathrm { C } _ { n }$ be the center of the circle inscribed in triangle $\mathrm { A } _ { n } \mathrm { OB } _ { n }$, and let $S _ { n }$ be the area of triangle $\mathrm { A } _ { n } \mathrm { OC } _ { n }$.\\
What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } } { n }$? [4 points]\\
(1) $\frac { 1 } { 12 }$\\
(2) $\frac { 1 } { 6 }$\\
(3) $\frac { 1 } { 4 }$\\
(4) $\frac { 1 } { 3 }$\\
(5) $\frac { 5 } { 12 }$