>>> series(solve(exp(-x**2/2) / x - 1/n, x)[0], n, oo, n=3)
n + O(n**(-3), (n, oo))
>>> series(solve(exp(x**2/2) * x - n, x)[0], n, oo, n=3)
n + O(n**(-3), (n, oo))

This is incorrect, the correct growth rate should obviously be O(sqrt log n).
For, notice that for f_1(n) = sqrt log n, we have exp(f_1(n)^2/2) * f_1(n) = exp(log n / 2) * sqrt(log n) = sqrt n * sqrt log n = o(n). In fact, a matching upper bound on x can also be derived, since f_2(n) = 2 sqrt log n gives omega(n), and the LHS in the equation is monotone.
My proof also agrees with simple experiments:

>>> nsolve(exp(x**2/2) * x - 10**6, x, 5)
4.94317756154226

(of course, this is not "n + O(n^{-3})")