Which is the function that this sequence of functions converges

Question

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

Answer 1

Hint: The limit function $f$, where it exists, should satisfy $(f(x))^2=x+f(x)$.

For existence, perhaps use the fact that an increasing sequence that is bounded above converges.

Answer 2

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 $ .