As the title says I would like to know if interesting pattern occurs starting form numerical analysis with the conitnued fraction :
$$f(x)=\frac{x}{x!+\frac{2x^{2}}{x!!+\frac{3x^{3}}{x!!!+\frac{4x^{4}}{x!!!!+\frac{5x^{5}}{x!!!!!+....}}}}}$$
$$x!=\Gamma(x+1),x!!=\Gamma(1+\Gamma(x+1)),\cdots$$
As the expansion increases the function seems to behave like $g(x)=x\sin\left(\frac{1}{x}\right)$ around $x=2$
Some various local extrema appears so let a picture :
How to explain this behavior around $x=2$ ? Do you some other interesting fact about $f(x)$ ?