Which is the function that this sequence of functions converges Prove that $$ \left(\sqrt x, \sqrt{x + \sqrt x},  \sqrt{x +  \sqrt {x + \sqrt x}}, \ldots\right)$$ in $[0,\infty)$ is convergent and I should find the limit function as well.
For give a idea, I was plotting the sequence and it's look like

Note that $f(x)$ is limited by $\sqrt{x}+1$ as can be shown easily by $induction$.
For the limit $y=f(x)$ we have the following :
$\sqrt{x+y}=y$
$x+y=y^2$
$0=y^2-y-x$
$\Delta=   1+4x                    $
$ So $  ,   $y=f(x)=\frac{1+\sqrt{1+4x}}{2}$  for $x>0$ and  $f(0)=0 $
.
