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Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

Tetration is defined as

\[\Large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \ a\text{'s}}.\]

Find the value of

\[\lim_{n\rightarrow\infty}{^{n}}\left(\sqrt {2}\right).\]

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\[\lim_{n\to\infty}\frac{P_{n-1} P_{n+1}}{P_{n}^2} = \]

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The sequence \(\{a_n\}\) follows the recursion \(a^2_{n+1}=2a_n+3\) with \(a_1=7.\)

Determine \(\displaystyle \lim_{n \to \infty} a_n\).

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