L’Hospital’s Rule

1. L’Hospital’s Rule Tiffany Hall and Amanda Boggs

2. L’Hospital • Born in Paris in 1661 • Was a cavalry officer until his nearsightedness forced his resignation • After he resigned, he studied math under Johann Bernoulli

3. L’Hospital and Bernoulli • L’Hospital agreed to support Bernoulli’s works if he would publish L’Hospital’s book • L’Hospital’s book, Analyse des infiniment petits pour l'intelligence des lignes courbes, contained L’Hospital’s Rule • There was a dispute as to whose ideas were used in the book. L’Hospital claimed they were his own, but Bernoulli stated that they were his. Letters written between the two mathematicians were found in 1955 which proved that Bernoulli’s claims were correct. Johann Bernoulli

4. L’Hospital’s Rule (according to Stewart Calculus 3rd Edition) “Suppose that f and g are differentiable and g’(x)  0 on an open interval I that contains a (except possibly at a). Suppose that: =0 and =0 or that and = = Then: = if the limit on the right side exists (or is + or - )”

5. Indeterminate Forms

6. How to Apply the Rule • If it’s in a fractional form, take the derivative of the top and the derivative of the bottom. Then take the limit. Repeat this process until the answer is not in an indeterminate form.

7. Example w/ the Fractional Form = 1 =

8. Proof Look at the linear approximation to f(x) and g(x) at x=a When x is close to a, the the ratio is: This ratio approaches f’(a)/g’(a) as x a is g’(a)  0

9. Works Cited • Stewart, James. Calculus. 3rd ed. Pacific Grove, CA: Brooks/Cole Publishing Company, • 1995. • http://www.geocities.com/mathladies/bios/johannb.html • http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html • http://www-math.mit.edu/~djk/18_01/chapter26/proof01.html