Prooving Integral converge

Question

We have a function $$f:[0;\infty) \to R$$ such that

$$\lim_{\substack{x\to \infty}} (\ln f)'(x)$$ exists and is negative.

I need to proove that the integral $$\int_0^\infty f(x) \mathrm{d}x$$ converges.

I fall in the trap that thinking if $f(x)$ is derivable so $f'$ will be integrable but it's not true ...

Answer 1

Let the limit of $(\ln\, f)'$ be $-l$ (so $l >0$). There exists $T>0$ such that $(\ln\, f)'<-l/2$ for $x \geq T$. Now $\ln\, f(x) = \ln\, f(T) +\int_T^{x} (\ln\, f)' <\ln\, f(T)-l/2 (x-T)$ for $x >T$. Hence $f(x) <f(T)e^{-l/2(x-T)}$ for $x >T$. Integrability of $f$ is immediate from this.