This is completely off-topic, but did you mean to call it l'hospital as opposed to L'Hôpital? I remember even my calculus book had similar errors, and I always wondered if it was just because the two looked so similar (or if there was any more reasoning behind it). didn't mean to nitpick, your comment just triggered a repressed train of thought :)
His original name was actually Guillaume de l'Hospital; French spelling reforms later did away with a number of cases of silent 's' (which had been silent for a long time already), replacing it with a circumflex over the preceding vowel.
ah, thanks - I had a feeling it was something like that! I figured a Calculus book would probably get it right :)
It's not an error. The french changed their spelling to replace a silent 's' after some letters with a circumflex over those letters. Using the silent s instead of a circumflex is considered correct when transcribing into English.
No. What would you apply it to in 0^0? The best you could do is come up with two functions, f(x) and g(x), with f(x) and g(x) going to 0 as x goes to 0, and try to use it to evaluate f(x)^g(x), but the result is going to depend on exactly what your choices are for f(x) and g(x).
A precondition of L'Hôpital's Rule is that the limit in question exists. So: Prove that the limit exists, and then you can bust out L'Hôpital's Rule to prove that it's 1.
That's addressed in the article under "cleverest student."
And the cleverest student turns out to be right, given the convention asserted by the mathematician. If 0⁰ were 0, or indeterminate, then the limit wouldn't exist and, therefore, L'Hôpital's Rule wouldn't apply. But given that it's 1, the limit does exist, and all is well until Zermelo-Fraenkel is proven inconsistent.