Let $x(n)$ be the solution to the following equation $$ x=-\frac{\log(x)}{n} \quad \quad \quad \quad (1) $$ as a function of $n,$ where $n \in \mathbb N.$ How would you find the asymptotic behaviour of the solution, i.e. a function $f$ of $n$ such that there exist constants $A,B$ and $n_0\in\mathbb N$ so that it holds $$Af(n) \leq x(n) \leq Bf(n)$$ for all $n > n_0$ ?
Asymptotics of a solution
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asymptotics
special-functions