Problem statement: I am trying to recover a solution of one of my courses.
Let assume we have a function y that writes :
$y (p,N) = \left(\frac{1}{2} - \frac{\gamma}{4p}\right)-\sqrt{\left(\frac{1}{2} - \frac{\gamma}{4p}\right)^2 -\frac{1}{2pN} } \ \ \ \ \ \ \ \ \ \ \ (1) $,
which is a function of the parameters: $ \left\{p \in \mathbb{R} | \ p \ne 0 \right\} $, $\gamma = \left(1-\frac{1}{N} \right)$ and $ \left\{N \in \mathbb{N} | \ N \ne 0 \right\}$.
In the large N limit, when $N>> 1$, the solution should read:
$y (p,N>> 1) = \frac{1}{(2p-1)N}. \ \ \ \ \ \ \ \ \ \ \ (2)$
Question: Maybe it is trivial but I have some difficulties to recover (2). I tried to just use the definition of $\gamma$, replace in (1), then keep only terms in $1/N$ but I did not find the correct solution. Thank you very much for any help on this !