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$\lim _{n \rightarrow x} \sqrt{2}\left[\frac{(2+\sqrt{2})^n+(2-\sqrt{2})^n}{(2+\sqrt{2})-(2-\sqrt{2})^n}\right]=$
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Verified Answer
The correct answer is:
$\sqrt{2}$
For $n \rightarrow \infty$
$\begin{aligned} & \because 2-\sqrt{2} < 1 \\ & \therefore \lim _{n \rightarrow \infty}(2-\sqrt{2})^n \rightarrow 0\end{aligned}$
$\begin{aligned} & =\lim _{n \rightarrow \infty} \sqrt{2}\left[\frac{(2+\sqrt{2})^n+0}{(2+\sqrt{2})^n-0}\right] \\ & =\sqrt{2}\end{aligned}$
$\begin{aligned} & \because 2-\sqrt{2} < 1 \\ & \therefore \lim _{n \rightarrow \infty}(2-\sqrt{2})^n \rightarrow 0\end{aligned}$
$\begin{aligned} & =\lim _{n \rightarrow \infty} \sqrt{2}\left[\frac{(2+\sqrt{2})^n+0}{(2+\sqrt{2})^n-0}\right] \\ & =\sqrt{2}\end{aligned}$
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