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?

Solution 1:

Hint: Consider $x\mapsto x^\alpha$, for an appropriate $\alpha$.

Solution 2:

Yes. For example, $f(x)=\dfrac{1}{\sqrt[4]x}$

$\lim\limits_{x\to\infty} \dfrac{x}{(\sqrt[4]{x})^3}=\infty$

$\lim\limits_{x\to\infty} \dfrac{x}{(\sqrt[4]x)^5}=0$

Solution 3:

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 $0$.