Just to restate my question as in the title:
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ , for some real $M$, imply that $f(x)\leq |x| M $ ?
Any help would be appreciated!
2026-04-13 04:27:23.1776054443
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ ,for some real $M$ ,imply that $f(x)\leq |x| M $?
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The inequality won't hold on any neighborhood of $0$. For instance, $$\lim_{x \to 0} \frac{x^2}{|x|} \le 0$$ but $x^2 \le 0|x|$ is false for all $x$ except $x=0$.