Let \(\left\{x_{n}ight\}_{n=1}^{\infty}\) be a sequence of real numbers defined by

\[x_{n}=\left\{\begin{array}{rl}-1 & n=1+3(k-1), k \in \mathbb{N} \\0 & n=2+3(k-1), k \in \mathbb{N} \\1 & n=3+3(k-1), k \in \mathbb{N}\end{array}ight. \]

Compute

\[\liminf _{n ightarrow \infty} x_{n}\]

and

\[\limsup _{n ightarrow \infty} x_{n}\]

Determine if the limit of \(x_{n}\) as \(n ightarrow \infty\) exists.