What 's equal this :$\lim_{x\to \infty} \operatorname{erf}(\operatorname{erf}(\operatorname{erf}(\cdots \operatorname{erf}((x)))))) $?

Question

Error function is a special function (non-elementary ) ,it is defined as follow :$$\operatorname{erf}(x)= \frac{1}{\sqrt{\pi}}\int_{-x}^x e^{-t^2}\ dt,$$ I would like to know what's equal this limit : $$\lim_{x\to \infty} \operatorname{erf}(\operatorname{erf}(\operatorname{erf}(\cdots \operatorname{erf}((x)))))). $$

Note: I know only this :$\lim_{x \to \infty} \operatorname{erf}(x)=1$ and $\operatorname{erf}(\operatorname{erf}(\operatorname{erf}(\cdots \operatorname{erf}((x)))))) $ is a composition of $\operatorname{erf}$ $n$-times.

Thank you for any help

Answer 1

One may use $\lim_{x \to \infty} \operatorname{erf}(x)=1$ and one may use the fact that $\operatorname{erf} (\cdot)$ is continuous over $\mathbb{R}$, then $$ \lim_{x\to \infty} \operatorname{erf} (\operatorname{erf} (\operatorname{erf} (\cdots \operatorname{erf} (x)\cdots)=\operatorname{erf} (\operatorname{erf} (\operatorname{erf} (\cdots \operatorname{erf} (1)\cdots) $$ where $\operatorname{erf}$ appears $n$ times on the left hand side and $n-1$ times on the right hand side.