Let \(\left\{A_{n}ight\}_{n=1}^{\infty}\) be a sequence of events from \(\mathcal{F}\), a \(\sigma\)-field on the sample space \(\Omega=\mathbb{R}\), defined by \(A_{n}=\left(-1-n^{-1}, 1+n^{-1}ight)\) for all \(n \in \mathbb{N}\). Compute

\[\begin{aligned}& \liminf _{n ightarrow \infty} A_{n}, \\& \limsup A_{n},\end{aligned}\]

and determine if the limit of the sequence \(\left\{A_{n}ight\}_{n=1}^{\infty}\) exists.