Consider the following implicit definition of $a$, $$\int_{a}^{a+\sqrt{\frac{a-b}{c}}} x \; p(x) \frac{(a-x)\left(a+\sqrt{\frac{a-b}{c}}-x \right)}{(x-b)(x-z)}dx = 0,$$
where $b<a<0<z$, $c>0$, and $p(x)$ is a probability density function with support large enough such that $p(x)>0$ for every $x\in\left[ a, a+\sqrt{\frac{a-b}{c}} \right]$. By using the implicit function theorem, I obtain that $$ \frac{d a}{d z}>0 .$$
I am interested in calculating the limit $$ \lim_{z \to +\infty} a(z). $$
In particular, I'd like to show if $a(z)$ gets arbitrarily close to $0^-$ as $z$ grows arbitrarily large, or instead $a(z)$ is bound away from zero at the limit.
To simplyfy the analysis, we can set $c=1$ and use as $p(x)$ the improper uniform over the real line so as to get rid of it.
Thanks a lot in advance.