TITLE: Can these two limits be true at the same time
QUESTION [12 upvotes]: 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?

REPLY [3 votes]: Assume $f(x) = x^P$, then to satisfy conditions listed,

$x^{5P + 1} \rightarrow 0 \Rightarrow 5P + 1 < 0$
$x^{3P + 1} \rightarrow \infty \Rightarrow 3P + 1 > 0$

So you get: $f(x) = x^P$, where $P \in \left]-\dfrac{1}{3}, -\dfrac{1}{5}\right[$