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?

If you look at $$L(n)=\lim_{x\to+\infty}xf^n(x)$$ as Git Gud suggested, use $f(x)=x^a$ and so you search for $$L(n)=\lim_{x\to+\infty}x^{an+1}$$ So, if $an+1 \gt 0$, the limit is $\infty$ and if $an+1 \lt 0$, the limit is $0$.
