Could there be function:

$\displaystyle f0,\infty) \rightarrow C$ so true: $\displaystyle \frac{1}{f} \in L^1(0,\infty)$ and $\displaystyle \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2}$ ?

Do I need to use some of inequalities or in any other way?

Thank you.