An iteration limit

Let $n \geq 1$ be a natural number and let $f_n=x^{x^{x^{\dots^{x}}}}$ where the number of $x$ ‘s in the definition of $f_n$ is $n$ . For example:

$f_1 = x \quad , \quad f_2 = x^x \quad , \quad f_3=x^{x^x}$

Evaluate the limit:

$\ell = \lim_{x\rightarrow 1} \frac{f_n(x) - f_{n-1}(x)}{\left ( 1-x \right )^{n+1}}$

Solution

We easily see by induction that

$f_n(x) = 1 +\mathcal{O} \left ( 1-x \right ) \quad \text{for} \quad n \geq 1$

as well as

$f_n(x) = f_{n-1}(x) + (-1)^n \left ( 1-x \right )^n + \mathcal{O} \left ( \left ( 1-x \right )^{n+1} \right ) \quad \text{for} \quad n \geq 2$

Thus,

$\frac{f_n(x) - f_{n-1}(x)}{\left ( 1-x \right )^n} = (-1)^n + \mathcal{O}\left ( \left ( 1-x \right )^{n} \right )$

and thus $\ell$ does not exist.