My question es exactly the one in the title. I am trying to find any smooth function $f\in C^\infty((1,\infty))$ such that $$ f(x)\xrightarrow{x\to+\infty}0 \qquad \hbox{but} \qquad x\log(x)f'(x)\xrightarrow{x\to+\infty}+\infty. $$ So far I've tried with $1/\log(x)$ and $1/\log(\log(x))$, etc, with no success. Now I am wondering if that kind of function can actually exists. A priory looks like a quite reasonable requirement, but surprisingly I cannot find any such function.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
If $x\log(x)f'(x)\to+\infty$ as $x\to+\infty$, then there exists $M$ such that $f'(x)\ge\frac{1}{x\log(x)}$ whenever $x>M$. Integrating gives $$f(x)\ge f(M)+\int_M^x\frac{dt}{t\log(t)}$$ whenever $x>M$. And this integral can be evaluated as $$\log\log (x)-\log\log(M)$$ which tends to $+\infty$ as $x\to+\infty$, contradicting $\lim_{x\to+\infty}f(x)=0$.
