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

\[x_{n}=\frac{n}{n+1}-\frac{n+1}{n},\]

for all \(n \in \mathbb{N}\). Compute

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

and

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

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