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

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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.

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