# If $\lim_{x \to\infty} f(x)$ exists and $\lim_{x \to\infty} x f'(x) = 0$, then is $\lim_{x \to\infty} f(x) = 0?$

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.

• HINT: If you add a constant to $f(x)$, does that change $f'(x)$? – Ted Shifrin Aug 15 at 3:00
• 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. – 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.