Answer by Lefteris Angelis:

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.

by http://pballew.net

How was L’Hopital’s rule discovered?

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