Given
$$r(t)=\frac{f(t)}{1-F(t)} \tag{Eq. 1}$$ where $$f(t)=\frac{dF}{dt} \tag{Eq. 2}$$
and the conditions:
$$\lim_{t\rightarrow \infty} r(t)=1 \tag{Eq. 3}$$ $$\lim_{t\rightarrow \infty} F(t)=1 \tag{Eq. 4}$$ $$\lim_{t\rightarrow \infty} f(t)=1-F(t) \tag{Eq. 5}$$
I can think of just one function $F$ satisfying these three conditions--the logistic function:
$$F(t)=\frac{1}{1+e^{-t}} \tag{Eq. 6}$$
(which can also be expressed $F(t)=r(t)$)
Is this is the only function satisfying these conditions? If so, is there a way to prove it?