Prove or disprove the following statements: (A) (5 marks) Suppose that $f ( z )$ is a complex analytic function in the punctured unit disk $0 < | z | < 1$ such that $\lim _ { n \longrightarrow \infty } f \left( \frac { 1 } { n } \right) = 0$ and $\lim _ { n \longrightarrow \infty } f \left( \frac { 2 } { 2 n - 1 } \right) = 1$, then there exists a positive integer $N > 0$ such that $\lim _ { z \rightarrow 0 } \left| z ^ { - N } f ( z ) \right| = \infty$. (B) (5 marks) There exists a non-zero entire function $f$ such that $f \left( e ^ { 2 \pi i e n ! } \right) = 0$ for all $n \geq 2025$.
Prove or disprove the following statements:\\
(A) (5 marks) Suppose that $f ( z )$ is a complex analytic function in the punctured unit disk $0 < | z | < 1$ such that $\lim _ { n \longrightarrow \infty } f \left( \frac { 1 } { n } \right) = 0$ and $\lim _ { n \longrightarrow \infty } f \left( \frac { 2 } { 2 n - 1 } \right) = 1$, then there exists a positive integer $N > 0$ such that $\lim _ { z \rightarrow 0 } \left| z ^ { - N } f ( z ) \right| = \infty$.\\
(B) (5 marks) There exists a non-zero entire function $f$ such that $f \left( e ^ { 2 \pi i e n ! } \right) = 0$ for all $n \geq 2025$.