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For my first time to make a new topic, I would like to focus on some concepts such as proofs and problems from Marquis l'Hospital's (also spelled as l'Hôpital) Analyse.

First, some historical background on L'hospital and Analyse des infiniment petits pour l'intlligence des courbes:

Guillaume Francois Antoine de L'hospital, marquis of St. Mesme, born in 1661, is the son of Anne-Alexandre de L'hospital and Elizabeth Gobelin. His aristocratic family had eminent service to the king since 1488. He served as a cavalry officer in the French army, but his near-sightedness caused him to resign. He then focused on mathematics, and upon meeting Johann Bernoulli in 1691, they came into a financial agreement that Bernoulli would teach him calculus. This instruction was quite exclusive since he regularly paid Johann "half a professor's salary" for his lecture on calculus and allowed L’hospital to use Bernoulli’s discoveries in any manner. After five years, he published his book, Analyse des infiniment. His book is the first publication that tackles the field of calculus and it also presented some essential concepts of differentiation. This provided the definitions of “variable quantities” (which may explain the concept of limits) as “those that continually increase or decrease” and the differential as the “infinitely small part whereby a variable quantity is increased or decreased. Moreover, he gave a postulate stating that a “curve may be considered as a polygon with infinitely many sides and infinitely small length…” (this may translate to tangent lines). He claimed his legitimate ownership of the ideas expressed in the book, but also acknowledged his gratitude to Leibniz and Bernoulli in the preface. Although his work is greatly influenced by Bernoulli, the book included ideas that L’hospital, himself, was able to conduct on his own. Upon L'hospital's death on 1704, Bernoulli argued that he owns the ideas in the book, but only a few believed in him. Bernoulli died in January 1, 1748 and in 1922, the discovery of Bernoulli's manuscript supported his claim that the ideas in L'hospital's book came from him. Due to the letters between the two, it has been established that Bernoulli greatly contributed to L’hospital’s work.

There is also some modern explanation or proofs to l'Hospital's Rule:


The figure suggests visually why l'Hospital's Rule might be true. The first graph shows two differentiable functions f and g, each of which approaches 0 as x approaches a. If we were to zoom in toward the point (a, 0), the graphs would start to look almost linear, as in the second graph, then their ratio would be [m1(x - a)]/[m2(x - a)] = m1/m2 which is the ratio of their derivatives. This also suggests that the limit of the ratio f(x) over g(x) as x approaches a is equal to the limit of the ratio f'(x) over g'(x) as x approaches a.

(Stewart, James. Calculus: Early Transcendentals. Sinagpore: Brooks/Cole, 2003.)

(I think there is also another method, but I am still trying to find my other resources from my piles of papers.)

Excercise: Compare the statements of l'Hosptial (post # 2) and this, and show that these are essentially the same.