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Let $f : (0, \infty) \rightarrow \mathbb{R}$ be differentiable and suppose that $|f(x)| \leq \frac{C}{x^k}$ ($k$ is non-negative) and this inequality would not hold for a smaller $k$ (even if you change $C$). Suppose this also holds for $|f'(x)|$ but with a possibly different $C$, but same $k$. Show that $\lim_{x \rightarrow 0^+} f(x)$ exists.

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    Could you be more specific about your statement : "this inequality would not hold for a smaller $k$" ? Does this mean $$\forall C'>0, \ \forall k'0, \ |f(x)| > C'/x^{k'}$$ ?2012-09-24
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    Yes, that's what I mean, vanna.2012-09-24
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    The whole elaborated setup is just a terribly fancy way to say that $k=0$. Where did the problem come from?2012-10-31
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    Is $k$ assumed to be integer? And is the claim that the limit is finite, or would $+\infty$ or $-\infty$ also be included in "the limit exists"?2012-10-31

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