Let (left{x_{n}ight}_{n=1}^{infty}) be a sequence of real numbers defined by (x_{n}=n 2^{-n}), for all (n in mathbb{N}).

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Let \(\left\{x_{n}ight\}_{n=1}^{\infty}\) be a sequence of real numbers defined by \(x_{n}=n 2^{-n}\), 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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