Suppose $\alpha\in(0,\frac12)$ and $\beta\in(0,\infty)$ are fixed. Initially I have $N\in\mathbb N\backslash\{0\}$ and $n\in\{0,\ldots,N\}$. I'd like to know, as a function $n$, the solution of the cubic equation
$ (N+2\beta)x^3-(N+n+3\beta)x^2+(n+\beta+N\alpha-N\alpha^2)x+n\alpha^2-n\alpha=0 $ in the limit $N\to\infty$.
Dividing the polynomial by $N$ gives $ (1+2\epsilon_2)x^3-(1+\epsilon_1+3\epsilon_2)x^2+(\epsilon_1+\epsilon_2+\alpha-\alpha^2)x+\epsilon_1(\alpha^2-\alpha) = 0, $ where $\epsilon_2 = \beta/N \to 0$. For fixed $n$, $\epsilon_1 = n/N \to 0$ so it seems in this case a multivariate generalization of perturbation theory might be appropriate. Does such a thing exist?
Bonus addition for the curious: the solution $q$ lies in $[\alpha,1-\alpha]$ and possess a symmetry about $\frac N2$. So the solution for $n = O(N)$ is given by the symmetry condition $q(n) = 1 - q(N-n)$. Also near $n=\frac N2$ the solution is well approximate by $q=\frac n N$.
Plotted below is the solution for $N=1001, \alpha = \frac12, \beta = \frac14$ in black and $q= \frac n N$ in red: