Counterexamples to l'hôpital's rule

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Counterexamples to L’Hôpital’s Rule R. P. Boas, Northwestern University, Evanston, IL60201 The American Mathematical Monthly,October 1986, Volume 93, Number 8, pp. 644–645. 1. Introduction.I am not, of course, claiming that L’Hôpital’s rule is wrong, merely that unless it is both stated and used very carefully it is capable of yielding spurious results. This is not a new observation, but it is often overlooked. For definiteness, let us consider the version of the rule that says that iffandgare differentiable in an intervalsa,bd, if limfsxd5limgsxd5, x®b®b` 2 2 x and if g9sxdÞ0in some intervalsc,bd, then limf9sxdyg9sxd5L 2 x®b implies that limfsxdygsxd5L. 2 x®b If limf9sxdyg9sxddoes not exist, we are not entitled to draw any conclusion about limfsxdygsxdspeaking, if. Strictlyg9has zeros in every left-hand neighborhood ofb, thenf9yg9is not defined onsa,bd, and we ought to say firmly that limf9yg9does not exist. There is, however, the insidious possibility thatf9andg9contain a common factor: f9sxd5ssxdcsxd,g9sxd5ssxdvsxd, wheresdoes not approach a limit and limcsxdyvsxd exists. It is then quite natural to cancel the factorssxd. This is just what we must not do in the present situation: it is quite possible that limcsxdyvsxdexists but limfsxdygsxd does not. This claim calls for an example. A number of textbooks give one, but it is (as far as I know) always the same example.

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