This may be a stupid question but I just want clarification about the use of the name of this rule. Well, most of the time what I see is L'Hospital's Rule, like in Baby Rudin and many other places. Occasionally, though, I do see what seems to be the "more correct" form L'Hôpital's Rule (like in Wikipedia). I believe the latter is the correct version, because the rule was indeed named after Guillaume de l'Hôpital. (see the wikipedia link).
But why is the former version so commonly used (much more than the latter I think)? The only reason I can imagine is the popularity of the English language. 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? In the first place, I think it is confusing to combine a French article with an English word, and in the second, it is even more confusing to use l' in place of le/la since in English the h in hospital isn't even mute. Anyway L'Hospital makes little sense to me..