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.

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.

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.