2. a) Give an example of two sequences of real numbers $\left(a_{n}\right)_{n \in \mathbb{N}}$ and $\left(b_{n}\right)_{n \in \mathbb{N}}$, such that $\lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=2$, but $\lim _{n \rightarrow \infty} a_{n} \neq \lim _{n \rightarrow \infty} b_{n}$.

b) Give an example of two sequences of real numbers $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ and $\left(b_{n}\right)_{n \in \mathbb{N}^{*}}$ such that $\lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=2$, but $\lim _{n \rightarrow \infty} a_{n}$ and $\lim _{n \rightarrow \infty} b_{n}$ do not exist.
2. a) For example, $a_{n}=2-\frac{1}{n}$ and $b_{n}=\frac{1}{n}$. It is demonstrated that they satisfy the conditions in the statement.

$3 p$

b) For example, $a_{n}=1-(-1)^{n}$ and $b_{n}=1+(-1)^{n}$. It is demonstrated that they satisfy the conditions in the statement.