# P boas in the american mathematical monthly c a b x g

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Chapter 0 / Exercise 144
College Algebra
Gustafson/Hughes
Expert Verified
L’Hôpital’s Rule” by R. P. Boas in TheAmerican Mathematical Monthly. c,a, b,xg xGUILLAUMEL’HÔPITAL(1661–1704)L’Hôpital’s Rule is named after the Frenchmathematician Guillaume François Antoine deL’Hôpital. L’Hôpital is credited with writingthe first text on differential calculus (in 1696)in which the rule publicly appeared. It wasrecently discovered that the rule and its proofwere written in a letter from John Bernoullito L’Hôpital. “… I acknowledge that I owevery much to the bright minds oftheBernoulli brothers. … I have made free use oftheir discoveries …,”said L’Hôpital.MathBioMathArticle
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Chapter 0 / Exercise 144
College Algebra
Gustafson/Hughes
Expert Verified
....SECTION 8.7Indeterminate Forms and L’Hôpital’s Rule569EXAMPLE 1Indeterminate Form Evaluate SolutionBecause direct substitution results in the indeterminate form you can apply L’Hôpital’s Rule as shown below.Apply L’Hôpital’s Rule.Differentiate numerator and denominator.Evaluate the limit.NOTEIn writing the string of equations in Example 1, you actually do not know that the firstlimit is equal to the second until you have shown that the second limit exists. In other words, ifthe second limit had not existed, it would not have been permissible to apply L’Hôpital’s Rule.Another form of L’Hôpital’s Rule states that if the limit of as approaches (or ) produces the indeterminate form or thenprovided the limit on the right exists.EXAMPLE 2Indeterminate Form Evaluate SolutionBecause direct substitution results in the indeterminate form youcan apply L’Hôpital’s Rule to obtainApply L’Hôpital’s Rule.Differentiate numerator and denominator.Evaluate the limit.0.limx1xlimxln xxlimxddxln xddxx,limxln xx./,0 0xf xg x2limx02e2x1limx0e2x1xlimx0ddxe2x1ddxxlimx0x0limx0e2x1xlimx0e2x100 0limx0e2x1x.0/0NOTETry graphing andin the same viewing window.Which function grows faster as approaches How is this observationrelated to Example 2??xy2xy1ln xlimxf xg xlimxfxg xTECHNOLOGYNumericaland Graphical ApproachesUse anumerical or a graphical approach toapproximate each limit.a.b.c.d.What pattern do you observe? Does ananalytic approach have an advantagefor these limits? If so, explain yourreasoning.limx052x1xlimx042x1xlimx032x1xlimx022x1xTry ItExploration AExploration BTry ItExploration AExploration B
..570CHAPTER 8Integration Techniques, L’Hôpital’s Rule, and Improper IntegralsOccasionally it is necessary to apply L’Hôpital’s Rule more than once to removean indeterminate form, as shown in Example 3.EXAMPLE 3Applying L’Hôpital’s Rule More Than OnceEvaluate SolutionBecause direct substitution results in the indeterminate form youcan apply L’Hôpital’s Rule.