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I have seen many questions about circular reasoning on this website but none of them have satisfied my doubt. How do I know if I'm doing circular reasoning on a limit by doing l'hôpital? I do know that circular reasoning is when you use l'hôpital on a limit like $\lim_{x\to 0}\frac{\sin(x)}{x}$ because the defn of the derivative of sine is cosine which is obtained by applying l'hôpital? Yeah but how do I know exactly if whenever I want to apply l'hôpital, that same l'hôpital has been used to evaluate the derivative?

Xetrez
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You appear to suggest at one point that you can use L'Hopital's rule to evaluate the derivative of a function. This is not true at all, and you don't even need to use the example of $$ \lim_{x \to 0}\frac{\sin x}{x} \, . $$ Consider the simpler case of how would you prove that the derivative of $x^2$ is $2x$. If your answer is 'use L'Hopital', then you would be using the fact that the derivative of $x^2$ is $2x$ in order to prove that the derivative of $x^2$ is $2x$ (a big problem!). L'Hôpital's rule is there to help you evaluate more complicated limits, where it has already been established what the derivative of all of the functions are. However, even in this case, it should be not the first tool that comes to your mind. Blindly applying L'Hôpital prevents people from thinking both analytically and creatively about the behaviour of functions. There is a little to be gained by using an overpowered tool, the proof of which is not trivial.

Joe
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  • Careful though: if you've already proved that $\sin' x=\cos x$ for all $x\ne 0$ and that $\cos$ is continuous at $0$, then technically you could use L'Hopital to prove that $\lim_{x\to0}\frac{\sin x}x=1$. Now, often this doesn't make too much sense, but there are instances of this being useful, namely in the theorem: if $f$ is differentiable on $(a-\varepsilon,a)\cup(a,a+\varepsilon)$, continuous at $a$ and $\lim_{x\to a}f'(x)=L\in\Bbb R$, then $f$ is differentiable at $a$ and $f'$ is continuous at $a$. –  Jan 18 '21 at 20:47
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    @Gae.S. That is true, and I think that's what makes it difficult to explain why it is wrong why to use L'Hôpital. Once you have actually proven that $\sin' = \cos$, you can use it, but it's just redundant. One other thing to note: say you mistakenly 'proved' that $\sin' = \tan$. Then, applying L'Hôpital to the limits won't show you that you've made an error. – Joe Jan 18 '21 at 20:48