Let $sigma in (0, 1)$ and $n geq1$ be fixed.

How can one show that the positive root $x_0 in (0, n)$ of

begin{equation}

f(x) = (n^2-x^2-sigma n^2 x)-(n^2-x^2+sigma n^2x)e^{-4x}

end{equation}

is of the form

begin{equation}

x_0 = x^*-varepsilon_n

end{equation}

for some $varepsilon_n$ such that $varepsilon_n rightarrow 0$ as $nto+infty$, and

where $$x^* := frac12(nsqrt{n^2sigma^2+4}-n^2sigma)$$ is the positive root of $f_1(x) = (n^2-x^2-sigma n^2x)$?