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This limit

[tex]\lim_{x \to 0} \frac{ \sin x}{x}[/tex]

is often cited as being an example where L'Hopital's rule cannot be used, since to use it you'd need to differentiate sine; but the derivative of sine, using the limit definition of a derivative, requires that you use the sinx/x limit (and the 1 - cosx / x limit) as part of the proof. So, we'd end up with a circular proof here, and thus we'd have to use the squeeze theorem as the alternative. But what struck me is if that is really the only proof that cosine is the derivative of sine? Aren't there others?

Apologies if this topic has been done to death.

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# Is it really circular to prove the sinx/x limit with L'Hopital?

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