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At the heart of it though, l'hopital's rule just seems to be a marriage of the ideas that differentiable functions are pretty darn close to their linear approximations at some point as long as you don't stray too far from that point and that for a continuous function, a small movement in the domain means a small movement in the value of the. How to prove l'hospital's rule for $\infty/\infty$ ask question asked 11 years, 5 months ago modified 2 years, 11 months ago 1 typically when they teach l'hopital's rule in school they just teach it algorithmically, that is just how to apply it, without the proof

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This is very similar to the way calculus in general is taught in most schools, i.e., just as a bunch of techniques, no proofs or justifications. Certainly there is a way to prove $\frac d {dx}\sin x=\cos x$ without using the said limit (if someone knows how, they can post it) so we don't even have any circular logic. L'hopital's rule is a local statement

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It concerns the behavior of functions near a particular point

The global issues (multivaluedness, branch cuts) are irrelevant. However, though hôpital does mean hospital in english, isn't it totally ridiculous to translate règle de l'hôpital into l'hospital's rule (just because the corresponding english word hospital happens to make sense) What's more, how are we supposed to pronounce l'hospital In an english way or in a french way?

16 a quick addition to ra1nmaster's otherwise excellent answer You can only apply l'hopital's rule if you have an indeterminate form and if the limit, after applying l'hopital's rule, exists. Many students learn l'hopital's rule and then forget how to use every other tool This is why after teaching l'hopital's rule, you should throw in a few examples where l'hopital's rule fails

This way, they think of l'hopital's rule as just another tool instead of magic.

Understanding the proof of l'hopital's rule ask question asked 8 years, 2 months ago modified 8 years, 2 months ago It should be used only when other simpler techniques (algebra of limits, squeeze theorem) fail And even when you really need to apply this rule, it is better to simplify the expression using algebra of limits and usual algebraic manipulation Jumping to l'hospital's rule for any and every limit problem is a bad bad bad idea.

The sine function fulfills the conditions of the l'hopital's rule Also, it is a fact that the derivative of sine is cosine, no matter how we proved it