Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 1 / a / Solution

Let $L=\log (1/ϵ)$. Taking logarithms gives $x^2+\log x=L$. Equivalently, $2x^2=W(2/{ϵ}^2)$ in terms of the Lambert W function. Iteration for large $L$ gives $x^2=L-\frac{1}{2}\log L+O((\log L)/L)$ and hence