real analysis – Root of a function as perturbation of polynomial

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 $$f_1(x) = (n^2-x^2-sigma n^2x)$$?