# Limits Indeterminate Forms and L Hospital s Rule

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1 Limits Indtrmint Forms nd L Hospitl s Rul I Indtrmint Form o th Tp W hv prviousl studid its with th indtrmint orm s shown in th ollowin mpls: Empl : Empl : tn [Not: W us th ivn it ] Empl : 8 h 8 [Not: W us th dinition h h 8 h o th drivtiv whr h h nd = 8] Empl : dinition o th drivtiv nd ] [Not: W us th whr

2 Howvr, thr is nrl, sstmtic mthod or dtrminin its with th indtrmint orm Suppos tht nd r dirntibl unctions t = nd tht is n indtrmint orm o th tp ; tht is, nd Sinc nd r dirntibl unctions t =, thn nd r continuous t = ; tht is, = nd = Furthrmor, c nd r dirntibl unctions t =, thn nd Thus, i, thn i nd r continuous t = This illustrts spcil cs o th tchniqu known s L Hospitl s Rul L Hospitl s Rul or Form Suppos tht nd r dirntibl unctions on n opn intrvl continin =, cpt possibl t =, nd tht nd I hs init it, or i this it is or, thn Morovr, this sttmnt is lso tru in th cs o it s,,, or s In th ollowin mpls, w will us th ollowin thr-stp procss: Stp Chck tht th it o is n indtrmint orm o tp I it is not, thn L Hospitl s Rul cnnot b usd Stp Dirntit nd sprtl [Not: Do not dirntit u th quotint rul!]

3 Stp Find th it o I this it is init,, or, thn it is qul to th it o I th it is n indtrmint orm o tp, thn simpli lbricll nd ppl L Hospitl s Rul in Empl : Empl : tn sc Empl : h 8 h h h 8 h h 8 h 8 Empl : Empl : twic] [Us L Hospitl s Rul Empl :, or whr Empl 7: [This it is not n indtrmint orm o th tp, so L Hospitl s Rul cnnot b usd]

4 II Indtrmint Form o th Tp W hv prviousl studid its with th indtrmint orm s shown in th ollowin mpls: Empl : Empl : Empl : it dos not ist Empl : bcus < nd thus =

5 Howvr, w could us nothr vrsion o L Hospitl s Rul L Hospitl s Rul or Form Suppos tht nd r dirntibl unctions on n opn intrvl continin =, cpt possibl t =, nd tht nd I hs init it, or i this it is or, thn Morovr, this sttmnt is lso tru in th cs o it s,,, or s Empl : 7 Empl : Empl : 8 Empl : 8 L Hospitl s Rul dos not hlp in this sitution W would ind th it s w did prviousl

6 Empl : = Empl : rctn Empl 7: rctn [This it is not n indtrmint orm o th tp, so L Hospitl s Rul cnnot b usd] III Indtrmint Form o th Tp Indtrmint orms o th tp cn somtims b vlutd b rwritin th product s quotint, nd thn pplin L Hospitl s Rul or th indtrmint orms o tp or Empl : Empl : csc csc cot tn tn Empl : [Lt ]

7 IV Indtrmint Form o th Tp A it problm tht lds to on o th prssions,,, is clld n indtrmint orm o tp Such its r indtrmint bcus th two trms rt conlictin inluncs on th prssion; on pushs it in th positiv dirction nd th othr pushs it in th ntiv dirction Howvr, its problms tht ld to on th prssions,,, r not indtrmint, c th two trms work tothr th irst two produc it o nd th lst two produc it o Indtrmint orms o th tp cn somtims b vlutd b combinin th trms nd mnipultin th rsult to produc n indtrmint orm o tp or Empl : Empl : V Indtrmint Forms o th Tps,, Limits o th orm or rquntl iv ris to indtrmint orms o th tps,, Ths indtrmint orms cn somtims b vlutd s ollows: 7

8 Th it on th rihthnd sid o th qution will usull b n indtrmint it o th tp Evlut this it u th tchniqu prviousl dscribd Assum tht = L Finll, L L L Empl : Find Lt This is n indtrmint orm o th tp Thus, Empl : Find This is n indtrmint orm o th tp Lt = Thus, = Empl : Find This is n indtrmint orm o th tp Lt tn Thus, = 8

9 Prctic Sht or L Hospitl s Rul [ ]

10 7 rctn 8 t dt rctn rctn rc tn

11 Solution K or L Hospitl s Rul [ ] Lt Now, lt Thus, =

12 Lt 7 Lt Thus, it dos not ist

13 Lt Thus, Lt Nt, lt Thus,

14 = 8 rctn dt t Lt Thus rctn tn sc rc tn rctn Lt Lt

15 Thus, Lt Thus,

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