For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points] (1) $\frac { 3 } { 2 }$ (2) $\frac { 5 } { 4 }$ (3) 1 (4) $\frac { 3 } { 4 }$ (5) $\frac { 1 } { 2 }$
For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant.\\
For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points]\\
(1) $\frac { 3 } { 2 }$\\
(2) $\frac { 5 } { 4 }$\\
(3) 1\\
(4) $\frac { 3 } { 4 }$\\
(5) $\frac { 1 } { 2 }$