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In class, a teacher made the following statement:

Whenever the limit of a real function of a real variable gives $\frac{0}{0}$ for $x \to a$, the limit can be calculated (via L'Hopital for example). I haven't seen any books claiming this (or giving a counterexample).

Does anyone know how to prove this statement or give a counterexample?

RobPratt
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Thiago Alexandre
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  • What does "can be calculated" mean? It's not true that such a limit always exists, if that's what you mean. – lulu May 24 '23 at 23:39
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    I think your teacher meant a possible way to calculate the limit is L'Hopital's rule. The limit of $\dfrac{f}{g}$, where $f,g\to 0$ as $x\to a$, does not necessarily exist, for instance $$\lim\limits_{x\to 0}\frac{x\sin\frac{1}{x}}{x}$$. And when it exists, maybe we are able to express its value with the symbols we have created (i.e. the analytic solution), and may be we can not. – Asigan May 24 '23 at 23:54
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    Its of course a given that $f,g\in C$ i.e. are at least once differentiable about $a$. Additionally, you can get stuck in endless loops as this post displays: https://math.stackexchange.com/questions/912650/when-does-l-hopitals-rule-fail – Henry Lee May 24 '23 at 23:59
  • That's correct Asigan and Henry Lee. This is elucidated for me. – Thiago Alexandre May 25 '23 at 10:34

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