Consider Cantor staircase function $f:\ [0,1]\rightarrow [0,1]$ and the moment function $F(x):=\int_0^1 (f(t))^x dt$. When $x$ tends to $+\infty$, it behaves like $x^{-\sigma}$, $\sigma:=\ln 3/\ln 2$. But the limit $F(x)\cdot x^{\sigma}$ does not exist: this value oscillates very slowly around a constant $1.9967\dots$

See more in Russian here: http://www.math.spbu.ru/analysis/f-doska/lap_can.pdf