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I'm currently working with limits in our Real Analysis course, and I'm currently struggling with certain implications regarding limits.

I can give you some examples:

  • If $f$ is bounded, then $f'$ is bounded

  • If $\lim_{x \rightarrow \infty} f(x) = \infty$, then $\lim_{x \rightarrow \infty} f'(x) = \infty$

  • If $\lim_{x \rightarrow \infty} f'(x) = \infty$, then $\lim_{x \rightarrow \infty} f(x) = \infty$

I understand the topics regarding continuity, and how differentiability implies continuity and stuff. Still, I've had a hard time to quickly figure out whether these implications are false or true. Not only the above mentioned, but implications of that nature in general.

I'd be glad if you could share any tips with you on how to think when confronted with problems as these.

As far as I know, it's always good to test the statement with some basic functions such as $\sin(x)$ or $\ln(x)$, but that can only be used to disprove them, and not to prove them.

Thank you.

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Tanamas
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1 Answers1

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Disprove an implication by a couterexample is indeed a good proof, usually a very effective way. With reference to you examples we have:

  • If $f$ is bounded, then $f'$ is bounded: $\sqrt x$ for $x\in[0,1]$

  • If $\lim_{x \rightarrow \infty} f(x) = \infty$, then $\lim_{x \rightarrow \infty} f'(x) = \infty$: $\ln x \to \infty$ but $(\ln x)'=\frac 1x \to 0$

To find them it's a good way try at first with elementary functions but there is not a general method to find them other than experience, immagination and creativity.

When we can't find a counterexample then we can take into consideration that the implication could be true and in this case we need a specific proof, as for the third example in the related

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