# $f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ asymptotic?

Consider for positive real $x$ :

$$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$$

How does this function behave ?

How fast does it grow ? Faster than any fixed iteration of exp sure, but How fast exactly ??

A brute estimate would be $\operatorname{tet}( \ln(x+1))$ but that would likely be a bad estimate. We have to do better.

• Could you use parentheses to make it clear which is a power of what? – Arnaud Mortier Feb 6 '18 at 18:17
• It is a square root ^ Cuberoot etc ... I failed the root symbols Tex... – mick Feb 6 '18 at 18:19
• So it's $$f(x) = (x^{1/2})^{(x^{1/3})^{(x^{1/4})^\cdots}}$$? – Antonio Vargas Feb 6 '18 at 18:23
• Yes indeed Antonio – mick Feb 6 '18 at 19:57