I am supposed to evaluate limit of a Lebesgue integral: $$\lim_{n \rightarrow \infty} \int_{[1, \infty]} \frac{\ln(nx)}{x + x^2\ln(n)} dm$$ It is, really, not that hard to determine the limit by the theorem about dominated convergence, if we know that $$\frac{\ln(x) + \ln(n)}{x + x^2\ln(n)} \le \frac1{x^2} + \frac{\ln(x)}{x^2}, \forall n \in \mathbb{N}.$$
I've tried proving that the sequence is decreasing and that the claim holds for $n = 1$. However, I couldn't prove that, because the function has a maximum at $1 - x\ln(x)= 0$, or it seems that way. After that point the function is decreasing and I couldn't determine the point of maximum as, apparently, the Lambert W function. I've also tried derivating (by $n$) left hand side subtracted by the right hand side, but I got the same. Am I doing something wrong? Any help is appreciated.