f$($s$,$x$)=\backslash$dfrac$\{2+3$s$\backslash$sqrt$[3]\{$x$\}\}\{(1+$s$\backslash$sqrt$[3]\{$x$\})(1+$x$)\}.$

Let s$\_0=0\mathrm{.}01,$s$\_\{$k$+1\}=\backslash$int$\_0^\{$s$\_$k$\}$f$($s$\_$k$,$x$)$dx$.$

Show that $\backslash$lim$\_\{$k$\backslash$to$\backslash$infty$\}$s$\_$k exists.

I can understand that ff behaves like $2$near $0$and behaves like $3\backslash$ln$\{$x$\}$when x is large enough.

I want to use Monotone Convergence Theorem, which needs to prove that ff is monotone and bounded.

However, I can't prove that the dividing point of ff being increasing, by acting like $2,$and being decreasing, by acting like $3\backslash$ln$\{$x$\},$is just the only fixed point s$^\backslash$ast$.$