$f(x) = x e^x$. Let $L : (0, \infty) \to (0, \infty)$ be its inverse function. Which of the following statements is correct?

Consider the function $f : (0, \infty) \to (0, \infty)$ given by $f(x) = x e^x$. Let $L : (0, \infty) \to (0, \infty)$ be its inverse function. Which of the following statements is correct?

1. $\lim_{x \to \infty}$$\frac{L(x)}{\log x} = 1.$

   2. $\lim_{x \to \infty} \frac{L(x)}{(\log x)^2} = 1.$

   3. $\lim_{x \to \infty} \frac{L(x)}{\sqrt{\log x}} = 1.$

   4. None of the remaining three options is correct.

I am unable to explicitly determine $L(x)$, the inverse of $f(x) = xe^x$. Hence, the remaining task is to approximate $L(x)$ asymptotically and analyse its growth in relation to functions of $\log x$ to resolve the given limits.