Suppose $f$ is a monotonic function such that $f(x)e^{f(x)} = x$. Find the value of $\int_0^e f$ and $\lim_{x \to \infty} \frac{f(x)}{\ln x}$

Question

I have a function $f : [0,\infty) \to \mathbb{R}$ such that :

• $f(x)e^{f(x)} = x$ for all $x \geq 0$.
• $f$ is monotonic.

• $\displaystyle\lim_{x \to \infty} f(x)=\infty $.

  Then, I have to evaluate : $$(1). \quad \int_0^e f(x) \mathrm{d}x$$ $$(2). \quad \lim_{x \to \infty} \frac{f(x)}{\ln x}$$

  Any suggestion?

Answer 1

Hint : Define a function $g:[0,\infty)\to [0,\infty)$ by the rule $$g(x)=xe^x$$ Now, you may to check that $g$ is non-decreasing over its entire domain (ckeck the sign of its derivative) and hence $g$ has an inverse. By the hypotheses of the problem we can conclude that $f$ is its inverse.

To solve $(1)$ substitute $u=f(x)$. Then $x=g(u)=ue^u$ and $dx=\dots$

Can you continue from here?

Note : $x=0$ iff $u=0$ and $x=e$ iff $u=1$.

Answer 2

With $w:=f(x)$, the integral is $\int_0^1(w+w^2)e^wdw=[(1-w+w^2)e^w]_0^1=e-1$, while the limit is $\lim_{w\to\infty}\frac{1}{1+\frac{\ln w}{w}}=\frac{1}{1+0}=1$.