Let $f(x)$ be a differentiable function on $(0, \infty)$ with $\lim_{x\to \infty} f(x) - xf'(x) = L\in \mathbb{R}$. I'm trying to prove or disprove that $\lim_{x\to\infty} f'(x)$ exists as well. Here's what I have so far: if the limit does exist, it must also equal $\lim_{x\to\infty} \frac{f(x)}{x}$ (divide the limit condition by $x$). Then I rewrote the condition as $x^2 \frac{d}{dx} \frac{f(x)}{x} \to L$, and from this I expect (using some $1/x^2$ asymptotic argument) my statement to be true - but I'm not sure how to rigorously proceed with this argument.. Can someone provide some next steps, or a counter example?
from Hot Weekly Questions - Mathematics Stack Exchange
Faraz Masroor
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