Zeroed Nested root

$\begin{aligned} \alpha &= \sqrt{d + \sqrt{d + \sqrt{d + \sqrt{d + \ldots}}}} \\ \alpha^2 &= d + \sqrt{d + \sqrt{d + \sqrt{d + \ldots}}} \\ \alpha^2 - d &= \sqrt{d + \sqrt{d + \sqrt{d + \sqrt{d + \ldots}}}} \\ \\ \\ &= \alpha \\ \\ \end{aligned}$

Rearranging,

$\begin{aligned} \\ \alpha^2 - \alpha - d &= 0 \\ \\ \alpha &= \dfrac{1 \pm \sqrt{1 + 4d} }{2} \\ \\ \end{aligned}$

We can discard the negative root as our nested radical is clearly positive. Then,

$\\ \displaystyle \lim_{d \to 0^+} \sqrt{d + \sqrt{d + \sqrt{d + \sqrt{d + \ldots}}}} = \displaystyle \lim_{d \to 0^+} \dfrac{1 + \sqrt{1 + 4d} }{2} = \dfrac{1 + \sqrt{1} }{2} = \boxed{1}$

We change $d$ to $x$ . We let $y$ denote the radical expression and graph the following $sqrt(x+y)=y$ . As x approaches 0 from the positive side, the expression approaches 1.