Consider the function $f(z)=1/z$ on the annulus $A=\{z \in \mathbb C : 1/2 < |z| < 2\}$. Which of the following is /are true?

Question

This is a question asked in my exam. I am unable to understand how to approach such problems.

Question: Consider the function $f(z)=1/z$ on the annulus $A=\{z \in\mathbb C : 1/2 < |z| < 2\}$. Which of the following is /are true?

$(1)~~$ There is a sequence $\{P_n(z)\}$ of polynomials that approximate $f(z)$ uniformly on compact subset of $A$.

$(2)~~$ There is a sequence $\{R_n(z)\}$ of rational functions, whose poles are contained in $\mathbb C\setminus A$ and which approximate $f(z)$ uniformly on compact subset of $A$

$(3)~~$ No sequence $\{P_n(z)\}$ of polynomials approximate $f(z)$ uniformly on compact subset of $A$.

$(4)~~$ No sequence $\{R_n(z)\}$ of rational functions whose poles are contained in $\mathbb C\setminus A$ approximate $f(z)$ uniformly on compact subset of $A$

Answer 1

1. It is false. If $\gamma\colon[0,2\pi]\longrightarrow\mathbb C$ is defined by $\gamma(t)=e^{it}$, then $\int_\gamma P(x)\,\mathrm dz$ for every polynomial function. Therefore, if there was such a sequence $(P_n)_{n\in\mathbb N}$ of polynomial functions, we would have $\int_\gamma\frac1z\mathrm dz=0$, which is not true.
2. Sure it's true. Just define $R_n(z)=\frac1z$ for each $n\in\mathbb N$.
3. True (see answer to question number 1).
4. False (see answer to question number 2).