MTH 251 – Differential Calculus Chapter 4 – Applications of DerivativesLimits of Quotients A ReviewSlide 3Slide 4Slide 5Limits of Quotients Indeterminate FormsLimits: Indeterminate FormsSlide 8L’Hôpital’s RuleL'Hôpital’s RuleL'Hôpital’s Rule: ExamplesL'Hôpital’s Rule: 0 - L'Hôpital’s Rule:  - L'Hôpital’s Rule: 00, 0, 1MTH 251 – Differential CalculusChapter 4 – Applications of DerivativesSection 4.6Indeterminate Forms andL’Hôpital’s RuleLimits of Quotients A Review)()(limxgxfaxCase 1:f(x)m and g(x)n as x a & m0, n0nmxgxfax)()(limCase 2:f(x)0 and g(x)n as xa & n00)()(lim xgxfaxLimits of Quotients A Review)()(limxgxfaxCase 4:f(x) and g(x) n as xa & n0Case 3:f(x)m and g(x)0 as x a & m0DNEor or )()(lim xgxfaxDNEor or )()(lim xgxfaxLimits of Quotients A Review)()(limxgxfaxCase 6:f(x) and g(x) 0 as xaCase 5:f(x)m and g(x) as xa & m00)()(lim xgxfaxDNEor or )()(lim xgxfaxLimits of Quotients A Review)()(limxgxfaxCase 7:f(x)0 and g(x) as xa0)()(lim xgxfaxLimits of Quotients Indeterminate Forms)()(limxgxfaxType 0/0: When f(x) 0 and g(x) 0 as xa.Type /: When f(x) and g(x) as xa.xxxx23lim000xxex3limLimits: Indeterminate FormsType 0 - :)()(lim xgxfax-When f(x) 0 and g(x) as xa.Type  - : )()(lim xgxfaxWhen f(x)  and g(x) as xa. [both + or both -]Types 00, 0, 1:)()(limxgaxxfWhen f(x) 0 and g(x)0 as xa.When f(x)  and g(x)0 as xa.When f(x) 1 and g(x) as xa.Limits: Indeterminate FormsType 0 - :Type  - :Types 00, 0, 1:xxxlntanlim00-111lim0xxexBoth + or both – at the same time. xxx3021lim1L’Hôpital’s Rule•First published in 1696In the 1st textbook on differential calculusWritten by Guillaume François Antoine, Marquis de l'Hôpital (1661-1704)First discovered by Johann Bernoulli•In 1694 they made a deal (revealed by Bernoulli in 1704 after l'Hôpital’s death). l'Hôpital paid Bernoulli 300 Francs a year to tell him of his discoveries, which l'Hôpital described in his book.•The story that l'Hôpital tried to get credit for inventing l'Hôpital's rule is false: he published his book anonymously, acknowledging Bernoulli's help in the introduction, and never claimed to be responsible for the rule.Pronunciation: lō-pē-’tälAlternate Spelling: l’HospitalL'Hôpital’s Rule•If f(x) & g(x) are differentiable over an open interval containing x = a, except possibly not at x = a, and …… is an indeterminate form of type 0/0 or /, then …)(')('lim)()(limxgxfxgxfaxax )()(limxgxfaxL'Hôpital’s Rule: Examples)(')('lim)()(limxgxfxgxfaxax If f(x)  0 & g(x)  0 or f(x)  ± & g(x)  ± as x  a, thenxxxx23lim0xxex3lim22lnlim4xxx�-L'Hôpital’s Rule: 0 - •Rewrite the limit as …•Example …)(')('lim)()(limxgxfxgxfaxax If f(x)  0 & g(x)  0 or f(x)  ± & g(x)  ± as x  a, thenxxxlntanlim0lim ( ) ( )x af x g x�where f(x)  0 & g(x)  ± as x  aOR1( )( )limx ag xf x�0/01( )( )limx af xg x�/L'Hôpital’s Rule:  - •Algebraically change f(x) – g(x) to get a quotient.Subtract fractionsMultiply numerator & denominator by a conjugateEtc……•Example …)(')('lim)()(limxgxfxgxfaxax If f(x)  0 & g(x)  0 or f(x)  ± & g(x)  ± as x  a, thenlim ( ) ( )x af x g x�-where f(x)  ± & g(x)  ± as x  a [both + or both –]111lim0xxexL'Hôpital’s Rule: 00, 0, 1•Take the natural log of both sides giving …•If this is equal to M, then … L = eM•Example …)(')('lim)()(limxgxfxgxfaxax If f(x)  0 & g(x)  0 or f(x)  ± & g(x)  ± as x  a, then( )lim ( )g xx aL f x�=where f(x)  0, , or 1 and g(x)  0 or  as x  agiving one of the above forms. xxx3021lim[ ]( ) ( )ln ln lim ( ) lim ln ( ) lim ( )ln ( )g x g xx a x a x aL f x f x g x f x� � �� �� �= = =� �� �0 - (–)0 -  -