For two points $\mathrm { P } ( n , f ( n ) )$ and $\mathrm { Q } ( n + 1 , f ( n + 1 ) )$ on the graph of the quadratic function $f ( x ) = 3 x ^ { 2 }$, let $a _ { n }$ be the distance between them. Find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$. (Here, $n$ is a natural number.) [4 points]
(1) 9
(2) 8
(3) 7
(4) 6
(5) 5

[Calculus] As shown in the figure, let Q be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point P on the circle meets the $x$-axis. For point $\mathrm { A } ( - 1,0 )$ and the origin O, let $\angle \mathrm { PAO } = \theta$. Find the value of $\lim _ { \theta \rightarrow \frac { \pi } { 4 } - 0 } \frac { \overline { \mathrm { PQ } } - \overline { \mathrm { OQ } } } { \theta - \frac { \pi } { 4 } }$. (where point P is in the first quadrant) [3 points]
(1) 2
(2) $\sqrt { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 1
(5) $\frac { \sqrt { 2 } } { 2 }$

On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP.
If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$. (Here, O is the origin, and $a , b$ are integers.) [3 points]