From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Limits/The rule of L'HOSPITAL... Date: 9 Jan 2000 20:01:17 -0500 Newsgroups: sci.math Summary: [missing] In article <387924CA.A4385411@math.columbia.edu>, Douglas Zare wrote: :Ronald Bruck wrote: :> In article <853uom$ma1@mcmail.cis.McMaster.CA>, :> kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) wrote: :> :> [about a misapplication of L'Hopital's Rule] :> ... :> : :> : FYI: mathematical rules worthy of the name have assumptions and :> :conclusions. And in many cases, the students are not to blame because :> :happy-go-lucky, assumptionless formulas and recipes are what they are :> :told to believe. The teachers or textbook authors may be great :> :"educators" but they may be clueless about what's a correct :> :mathematical statement. :> ... :> :> I don't think it's the textbooks or the faculty who inculcate this :> attitude in students, at least, not intentionally. And if unintentionally, it's a testimony to the authors'/instructors' flawed preparation. I happened to teach from such a textbook: they claimed (abbreviated) that if f->0 and g->0 (or both to infinity) as x ->infinity then lim(f(x)/g(x)) = lim(f'(x)/g'(x)) --- a statement easily refuted by counterexamples, one of them seen right in this thread, after an easy change of variable. This was in the sixth edition. Then someone might have pointed to them that there was something wrong with it, so in the seventh edition they played it safe: they dropped L'Hospital's Rule altogether. :> Everyone I know :> discourages this sort of routine (mis)application. And reading between :> the lines of the original poster, HIS instructor was doing this too; he :> gave two problems, immediately after covering L'Hopital's Rule, where :> it's not needed and/or doesn't work! : :In L'Hopital's rule, there is the blatant assumption that one needs to :have an indeterminate form, and there is the subtle assumption that the :denominator's derivative can't vanish arbitrarily close to the limit. I :mentioned the minor assumption but never checked it in any example, which :I think was reasonable; [...] Time to be technical: if a provable version (of the many versions I have seen) of L'Hospital's rule is presented for use then the requirement "the denominator is different from zero all over a sufficiently small deleted neighbourhood of the approached point" pops out of the other assumptions automatically and need not be checked at all. (Incidentally, Rudin seems to leave this observation to the students - it comes from Darboux's theorem that a function which is the derivative on an open interval possesses the Intermediate Value Property, whether it is continuous or not -- it cannot change sign without being zero in between.) A provable version of the "easy" (0/0) rule -- and only the x->c+ part will be formulated; extensions are obvious: Suppose g(x) -> 0, f(x) -> 0 , and suppose f'(x) / g'(x) -> L. (The derivatives are assumed to exist in a right deleted neighbourhood of c). Then f(x) / g(x) -> L . Note the logical structure: the imperative is that we try (not knowing whether we succeed) the quotient of the derivatives first; in other words, we do not steamroll through a recipe by boldly equating the two limits, and then cursing the mathematicians if the quotient of the derivatives fails to have a limit, or the latter limit problem turns out to be more difficult, while the misused formulations seem to promise the existence. An educational example (you've seen it, perhaps sterilized to save time): as x -> 0+ , try lim(x * log(x)) If you are out of luck, you try the 0/0 form, namely lim(x / (1/log(x)) and (carry it out) the resulting limit problem is harder than the original. Only then I tell the students not to despair but to try the infinity/infinity version - which, as we know, works here. :It is interesting how many of my students gave no indication whatsoever :that they learned anything in my course. A cluster of students showed :exactly the level of understanding I would expect from average students :after passing an AP Calculus AB class. Perhaps in their next class, they :will display no understanding of that material, but will seem to :understand the ideas I tried to teach them. [...] There is another phenomenon: the students are often conditioned to compartmentalize their knowledge: they don't want to relate the geography of France to the geography of Switzerland - never mind that these two countries share the Alps. Similarly, the students give me mean or helplessly blank looks if I come up with linear algebra terms and facts while teaching the parts of Real Analysis that discuss functions of many variables. Cheers, ZVK(Slavek).