Suppose $f(x)$ is increasing in $x.$ If $\displaystyle\lim_{x \to\infty} f(x)$ exists and $\displaystyle\lim_{x \to\infty} x f'(x) = 0$, then is $\displaystyle\lim_{x \to\infty} f(x) = 0?$

I think that it is true because $f'(x)$ decreases faster than $\dfrac{1}{x}$, but I cannot prove it.

  • 8
    $\begingroup$ HINT: If you add a constant to $f(x)$, does that change $f'(x)$? $\endgroup$ – Ted Shifrin Aug 15 at 3:00
  • $\begingroup$ If $f(x)$ is increasing in $x$ then $f'(x) \geq 0$. The fact that $\lim_{x \to\infty} x f'(x)=0$ would imply that $f'(x)$ would be zero somewhere. The derivative of a constant is zero. $\endgroup$ – Axion004 Aug 15 at 3:11

I found that the answer is no. An example is $f(x) = c - \frac{1}{x}$ for any constant c.


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