Infinite Power Towers

Calculus Level 2

Obviously, \[1^{1^{1^{.^{.^{.}}}}}=1\]

Furthermore, with a little bit of effort, we can show that \[\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}}=2\]

However, clearly \[2^{2^{2^{.^{.^{.}}}}}=\infty\]

So what is the largest real number \(x\) such that \(x^{x^{x^{.^{.^{.}}}}}\) is a finite number?

\[\frac{\pi}{2}\] \[e^{\frac{1}{e}}\] \[\frac{13}{9}\] \[\frac{3}{2}\] \[\sqrt[3]{3}\] \[\sqrt{2}\] \[\pi^{\frac{1}{\pi}}\] \[{\left(\frac{\pi}{e}\right)}^e\]

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