# L Hospital's Rule Essay

The Marquis de L’Hospital

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"I will be happy to give you a retainer of 300 pounds, beginning with the first of January of this year ... I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out ... I am not so unreasonable as to demand in return all of your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you not to disclose any of them to others. I ask you even not to send here to Mr. Varignon or to others any copies of the writings you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this ..."5 This excerpt from the letter shows that Bernoulli would be paid a generous amount in return for L’Hospital having the rights for his

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Bernoulli received no credit for the new material L’Hospital included, and in frustration, he wrote to many others, admitting to teaching L’Hospital the concepts that were written about in his textbook. Many were doubtful of Bernoulli’s claims because of the timing of his confession, but he was proved right through the 87 letters exchanged between Bernoulli and L’Hospital from 1692 to 17075. These letters showed that Bernoulli did indeed teach L’Hospital a significant amount of the principles L’Hospital had claimed as his own. L’Hospital’s original statement of his rule in his textbook was formulated geometrically with the use of a curve. The curve AMD, with AP=x, PM=y, AB=a, allows for y to be expressed as a fraction whose numerator and denominator become zero when x=a, meaning when P corresponds to B. The value of BD must be found. L’Hospital then uses examples, one with the formula . This equation is evaluated for when x=a. He uses the quotient of the differentials and sets x=a. The second example used is the expression . He solves without the new rule and instead solves for the radical and dividing by