Let $f : [1, \infty) \to [1, \infty)$ be a function such that $f(x) = x(1 + \ln x)$. Prove that $f$ is bijective and then calculate: $$\lim_{x \to \infty} \frac{f^{-1}(x) \ln x}{x}$$  I have no difficulties in proving that $f$ is bijective, but I can't calculate the limit. I've tried using l'Hospital's rule but got nothing meaningful.  Thank you in advance!
For any real number $a \geq 1$ let $f(a)$ denote the real solution of the equation $x(1+\ln x)=a$ then the    question is to find out $$ \lim_{a \to \infty} \frac{f(a)\ln a}{a}$$.  It is clear that if we denote $h(a)$ by $h(a)=a(1+\ln a)$ then $f(a)$ is the inverse function of $h(a)$. Also $f(a)$ is increasing function in its domain. Also the limit persuades using lhospital's but I cannot see how to apply it here. Thanks.
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