TITLE: Find functions $f:\mathbb{R}\to[0,1]$ such that: $\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1$
QUESTION [6 upvotes]: Find functions $f:\mathbb{R}\to[0,1]$ that satisfy: $$\lim_{x \to +\infty} \frac{f(x)^2}{f(2 x)}=1,$$
$$f'(x)\leq 0 \, \forall x,$$
and
$$\lim_{x \to +\infty} f(x)=0.$$
$$\lim_{x \to -\infty} f(x)=1.$$
I tried to find one without success.

REPLY [2 votes]: How about this:
$f(x) = \frac{1}{1+\text{exp}(x)}$?