This is a simplified version of a question I am looking at, but embarrassingly I can't do this one anyway. 

Let's assume $f(x)$ is a decreasing positive function, $f(0)$ is infinite and, moreover, $f(x)$ is comparable to $1/x$ near $x=0$. We look only at positive $x$'s.

**Question**. Assume we know that the limit
$$
\lim_{a\to 0^+}\int\limits_{a}^{2a} f(t)dt
$$
exists as a positive finite number. Does it imply that
$$
\lim_{n\to\infty} f(2^{-n})\cdot 2^{-n}
$$
exists? 

**Note**: it definitely does not imply that the limit of $xf(x)$ exists.