Tower of sqrt(2)

What is the limit of this infinite exponential?

Solution 1:

$$x=\sqrt{2}^x$$

This leads to x=2 or 4

Solution 2:

$$x=[\sqrt{2}^\sqrt{2}]^x$$

This leads to x not equal to 2 or 4

Solution 3: MatLab simulation of series

$$\sqrt{2}, \sqrt{2}^\sqrt{2}, [\sqrt{2}^\sqrt{2}]^\sqrt{2},…$$

leads to infinity

Solution 4: MatLab simulation of series

$$\sqrt{2}, \sqrt{2}^\sqrt{2}, \sqrt{2}^{[\sqrt{2}^\sqrt{2}]},…$$

leads to 2

Then what’s is value of this tower of sqrt(2)?

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