Tail equivalence for heavy-tailed data

Is it possible for a distribution $F(x)$ that has not a Pareto ($G(x)$) right tail equivalence to fit well heavy-tailed data? That is, to have ${\lim_{x\rightarrow\infty}}\frac{1-F(x)}{1-G(x)}=0$ instead of ${\lim_{x\rightarrow\infty}}\frac{1-F(x)}{1-G(x)}>0$