Can these two limits be true at the same time Is it possible to have $$\lim_{x\to+\infty}x\left[f(x)\right]^5=0$$ and $$\lim_{x\to+\infty}x\left[f(x)\right]^3=+\infty$$ for some function $f$ at the same time?

Yes. For example, $f(x)=\dfrac{1}{\sqrt[4]x}$
$\lim\limits_{x\to\infty} \dfrac{x}{(\sqrt[4]{x})^3}=\infty$
$\lim\limits_{x\to\infty} \dfrac{x}{(\sqrt[4]x)^5}=0$
