Wednesday, 14 May 2008

l'Hospital's Rule





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In 1694, Johann Bernoulli wrote a letter to Guillaume-Francois-Antoine de l'Hospital that included the theorem now known as l'Hospital's Rule (The alterante spelling "l'Hôpital" is often used in France) . There is little doubt for most math historians that a) Bernoulli first discovered it, and b) that L'Hospital first published it. l'Hospital is sometimes discredited because he published someone else's theorem, and paid for the privilege. My calulus teacher in college would rant about l'Hospital being an "inept" mathematician and "buying his fame", and gave me the impression that he had published it as if it were his own creation. The truth is, in the 1696 differential calculus book in which he published the theorem, L'Hospital thanks the Bernoulli brothers for their assistance and their discoveries. And in addition, he was far from inept as a mathematician. The MacTutor History of Math site comments that, "L'Hôpital was a very competent mathematician and solved thebrachystochrone problem."

L'Hospital never called the rule by his own name, and in fact, it appears that noone else did for several hundred years. Jeff Miller's web page on the first use of mathematical terms gives the first citation for the use as "de l'Hospital's theorem on indeterminate forms is found in approximately 1904 in the E. R. Hedrick translation of volume I of A Course in Mathematical Analysis by Edouard Goursat. The translation carries the date 1904, although a footnote references a work dated 1905 "

The rule is a method for finding the limiting behavior of a rational function whose numerator and denominator tend to zero at a point.(The rule is somewhat expanded today from its original form and can be used if both functions diverge to infinity also) In a traditional Calculus course, a student might use the rule to find, for example, the limit of the function as the value of x approaches 2. This meets the conditions since both the numerator and denominator approach zero as the value of x approaches two. The rule states that in such cases, one can take the derivative of the numerator and denominator independently and then find the limit of this ratio. Since the derivative of the numerator is 2x, and the derivative of the denominator is 1, the ratio of the derivatives is the experession 2x/1 or just 2x; and the limit as x approaches two for this function is just 2(2) = 4. In the area NEAR x=2, the value of the original ratio is very near 4.


The Stolz-Cesaro Theorem

Until recently I had never heard of this discrete analogue of l'Hospital's rule, and I thank the folks at Topological Musings blog for the lesson. The adjusted rule can apply to sequences (l'Hospital's rule is for continuous functions) under certain circumstances, and allow us to calculate the limit of the ratio of two divergent (they both go to infinity) seqences. If we think of the function above as a sequence in which the numerator ( x2-4) diverges to infinity as x grows larger and larger, and likewise the denominator (x-2) also grows without bound as x goes toward infinity, then the Stolz-Cesaro theorem says that .

So for out example, we need to find ((n+1)2 -4) - (n2-4) for the numerator, and ((n+1)-2)-(n-2) for the denominator. The numerator simplifies to (2n+1) and the denominator to (2n+2).. since these both still meet the conditions of the theorem, we can apply it once more to get 4/2 = 2... Using l'Hosptials rule for the same function as x-> infinity, we get (2x/x) and applying it once more we get two by l'Hospital as well.... (nice if they both get the same limit)..

The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro.

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