Let $f$ is continuously differentiable positive function, $\lim_{x\rightarrow\infty}f(x)=0$, and $\lim_{x\rightarrow\infty}f'(x)$ exist, is it true that $\lim_{x\rightarrow\infty}\frac{f'(x)}{f(x)}=0$.
My answer was yes, and this is my "proof".
Since the limit of $f'(x)$ exists, we can conclude that $\lim_{x\rightarrow\infty}f'(x)=0$. Suppose that $\lim_{x\rightarrow\infty}\frac{f'(x)}{f(x)}>0$. Hence, there are $c,k\in\mathbb{R}^+$ such that for every $x>k$, we have $\frac{f'(x)}{f(x)}>c$, or $f'(x)>cf(x)$. By letting $x\rightarrow\infty$, we shall get $0>0$, which is a contradiction.
However, my Sensei told me that this proof is wrong, but he didn't tell me which part. He just gave me a counter example that if $f=\phi$ (pdf of standard normal distribution), then we shall get the limit is infinity.
So, my questions are:
- which part of my proof is wrong?
- to make the conclusion true (the limit of such fraction is $0$), is there any additional premise I need?
- how about $\lim_{x\rightarrow\infty}\frac{[f'(x)]^2}{F(x)}$, where $F$ is the antiderivative of $f$? (I think if I can solve the problem of the second question, I'll get some ideas for the third)
Thank you so much in advance.
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