Find the number of real solutions to the equation $x = 99 \sin ( \pi x )$.

For a natural number $n$, let $a _ { n }$ be the $n$-th smallest $x$-coordinate among the intersection points of the line $y = n$ and the graph of the function $y = \tan x$ in the first quadrant.
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$? [4 points]
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { 3 } { 4 } \pi$
(4) $\pi$
(5) $\frac { 5 } { 4 } \pi$

The number of zeros of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 6 } \right)$ on $[ 0, \pi ]$ is $\_\_\_\_$.

Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$
The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \frac { \sin x } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) } + \frac { 2 } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) }$$
Then the number of solutions of $f ( x ) = 0$ in $\mathbb { R }$ is $\_\_\_\_$ .