limit with nested power of $x$

Question

Evaluation of $$\lim_{x\rightarrow 1}\frac{x^{x^{x^{x^{x^{x}}}}}-x^{x^{x^{x^x}}}}{(x-1)^6}$$

Try:: I try to solve it using Binomial Index for fraction

$\displaystyle x^x=\bigg[1+(x-1)\bigg]^x=1+x(x-1)+\frac{x(x-1)^2}{2}+\cdots $

Did not know how can i solve it,

Could some help me , thanks

Just a bit curious where such an expression arises. Which kind of application? — mathreadler, Mar 02 '19 at 11:08
l'Hopital may be a possibility, but it's not going to be pretty. — Arthur, Mar 02 '19 at 11:11

Robert Z · Accepted Answer · 2019-03-02T13:24:34.763

1

Let $t=x-1\to 0$ then $$x^s-1=e^{s\ln(1+t)}=st+O(st^2)$$ Hence $$x^x-x=x(x^{x-1}-1)=(1+t)(t^2+O(t^3))=t^2+O(t^3)$$ and
$$x^{x^x}-x^x=x^x(x^{x^x-x}-1)=(1+O(t))\left( (t^2+O(t^3))t+O((t^2+O(t^3))t^2)\right)=t^3+O(t^4).$$ In the same way, we get $$\frac{x^{x^{x^{x^{x^{x}}}}}-x^{x^{x^{x^x}}}}{(x-1)^6}=\frac{t^6+O(t^7)}{t^6}\to 1.$$

edited Mar 02 '19 at 13:24

answered Mar 02 '19 at 13:10

Robert Z

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limit with nested power of $x$

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