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Lesson 16 indeterminate forms (l'hopital's rule)
- 2. OBJECTIVES: • define, determine, enumerate the different indeterminate forms of functions; • apply the theorems on differentiation in evaluating limits of indeterminate forms of functions using L’Hopital’s Rule.
- 3. . ( )( ) ( ) 2313-xlim 1x 3x1-x lim 1x 3x4x lim :followsasnumeratorthefactorweexist,tolimit theforandform,ateindeterminanislimitthe 0 0 11 3)1(4)1( 1x 3x4x lim 1x 3x4x limofitlimtheEvaluate:callRe 1x1x 2 1x 22 1x 2 1x −=−== − − = − +− = − +− = − +− − +− →→→ → → 2 1x 3x4x lim,thus 2 1x −= − +− → used.bewillRulesHopital'L'onTheoremslimit saidtheevaluateToexample.secondthetoappliedbelonger nocanproblemsprevioustheinappliedprincipletheObviously, 0 0 0 )0sin( )0(2 )0(2sin 2x 2xsin lim 2x 2xsin limtheevaluatingconsiderusLet 0x 0x ===→ →
- 5. . Theorem 3.6.1 (p. 220) L'Hôpital's Rule for Form 0/0
- 6. . Applying L'Hôpital's Rule (p. 220)
- 7. Theorem 3.6.2 (p. 222) L'Hopital's Rule for Form ∞/∞
- 8. . 2x 2xsin lim.1 0x→ EXAMPLE: Evaluate the following limits. ( ) ( ) ( ) 0 0 0 0sin 02 02sin 2x 2xsin lim0x ===→ ( ) ( ) ( ) ( ) ( ) 10cos 2 02cos2 12 2x2cos lim 2x dx d 2xsin dx d lim 2x 2xsin lim :Rules'Hopital'LgsinuBy 0x0x0x === == →→→ 1 2x 2xsin lim 0x =∴ →
- 9. .3ysin-y 3y-ytan lim.2 0y→ ( ) ( ) ( ) ( ) 0 0 00 00 0sin3-0 03-0tan 3ysin-y 3y-ytan lim0y = − − ==→ ( ) ( ) ( ) ( ) ( ) 1 2 2 31 31 03cos-1 30sec 33ycos-1 13ysec lim 3ysin-y dx d 3y-ytan dx d lim 3ysin-y 3y-ytan lim :LHRBy 2 2 0y0y0y = − − = − − = − = − == →→→ 1 3ysin-y 3y-ytan lim 0y =∴ →
- 10. . ( ) ( )2 4 x x4 2xsinln lim.3 −ππ → ( ) ( ) ( ) ( ) ( ) ( )( )4x42 2x2cos 2xsin 1 lim x4 dx d 2xsinln dx d lim x4 2xsinln lim :LHRBy 4 x2 4 x 2 4 x −−π = −π = −π π → π → π → ( ) ( ) .ateminerdetinstillisThis 0 0 08 2 2cot 4 48 4 2cot2 x48 2x2cot lim 4 x − π = π −π− π = −π−π → ( ) ( ) 0 0 0 2 sinln 4 4 4 2sinln x4 2xsinln lim 22 4 x = π = π −π π = −ππ → ( ) 01ln :Note = ( ) ∞=∞ln ( ) −∞=0ln
- 11. . [ ] ( )[ ] ( ) 32 x2csc4 lim )4(8 2x2csc2 lim x48 dx d 2xcot2 dx d lim :LHRpeatRe 2 4 x 2 4 x 4 x − = −− − = −π− π → π → π → ( ) 8 1 1 8 1 4 2csc 8 1 x2csc 8 1 lim 2 2 2 4 x −=−= π −=−⇒ π → ( ) ( ) 8 1 x4 sin2xln lim 2 4 x −= − ∴ → ππ
- 12. . x 2 x e x lim.4 +∞→ ( ) ∞+ ∞ = ∞+ =⇒ ∞++∞→ ee x lim 2 x 2 x [ ] [ ] ( ) ( ) ∞+ ∞+ = ∞+ === ∞++∞→+∞→+∞→ e 2 1e 2x lim e dx d x dx d lim e x lim :LHRBy xx x 2 xx 2 x [ ] [ ] ( ) ( ) 0 2 e 2 1e 12 lim e dx d 2x dx d lim :LHRpeatRe xx x x = ∞+ ====⇒ ∞++∞→+∞→ 0 e x lim x 2 x =∴ +∞→
- 13. . 3xtanln 3xcosln lim.5 6 x π → ( ) ( ) ∞ ∞ = ∞ = π π = π π =⇒ π → - ln 0ln 2 tanln 2 cosln 6 3tanln 6 3cosln 3xtanln 3xcosln lim 6 x ( ) 01ln :Note = ( ) ∞=∞ln ( ) −∞=0ln [ ] [ ] ( ) ( )3x3sec x3tan 1 3x3sin cos3x 1 lim 3xtanln dx d 3xcosln dx d lim 3xtanln 3xcosln lim :LHRApply 2 6 x 6 x 6 x − == π → π → π → x3cos 1 3xcos x3sin lim x3sec x3tan lim 3x3sec x3tan 1 3xtan3 lim 2 2 2 6 x 2 2 6 x26 x π → π → π → −= −= • − ( ) 1 6 3sinx3sinlim 2 2 6 x −= π −=⇒ π → 1 3xtanln 3xcosln lim 6 x −=∴ → π
- 14. . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) applies.RulesHopital'L'casetheof eitherIn.or 0 0 toresultmaywhichevaluatedislimitthethen xg 1 xf limxgxflim ,Henceone.equivalentantodtransforme isproductstheirlimit,suchevaluateTolimit.itsapproaches xas0or0formthehavingundefinedisxgandxf ofproducttheunsigned),orsignedbecould(which0xglimand 0xflimthatsuchfunctionsabledifferentitwoarexgandxfIf.A :onDefininiti axax ax ax ∞ ∞ =• •∞∞• = = →→ → → ∞∞∞• -and0FORMSATEINDETERMINThe
- 15. . ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) Rule.sHopital'L'applyThen.or 0 0 toresultmay evaluatedwhenlimitwhosequotientequivalentaninto differencethengtransformibyevaluatedbecouldlimitThe .xglimxflimxgxflimisThat.- formtheofateindeterminbetosaidisxgxflimthe then,positivebotharewhichxglimand,xflimIf.B axaxax ax axax ∞ ∞ ∞−∞=−=−⇒∞∞ − ∞=∞= →→→ → →→ ∞∞∞• -and0FORMSATEINDETERMINThe
- 16. . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) LHR.thenandlogarithm ofpropertiestheapplythenfunction,theforyvariable alettingbyevaluatedbemayformsateindeterminThese ly.respective,1,,0formsateminerdetintheassumed xflimexpressionthethenorapproachesxasor xglimand,1xflim or,0xglimand,xflim or,0xglimand,0xflim :ifand,xgandxffunctionstwoGiven :Definition 00 xg ax axax axax axax ∞ → →→ →→ →→ ∞ ∞−∞+ ∞==• =∞=• ==• ∞ ∞ 1and,,0FORMSATEINDETERMINThe 00
- 17. . EXAMPLE: Evaluate the following limits: [ ]2xcscxlim.1 0x→ [ ] [ ] ∞•==→ 00csc02xcscxlim0x [ ] 0 0 0sin 0 sin2x x lim2xcscxlim :functionrationalequivalentantofunctionthegminTransfor 0x0x === →→ [ ] [ ] ( )( ) [ ] 2 1 2xcscxlim 2 1 )0(cos2 1 2cos2x 1 lim 2cos2x 1 lim sin2x dx d x dx d lim sin2x x lim :LHRApply 0x 0x0x0x0x =∴⇒== === → →→→→
- 18. . [ ]xlnxlim.2 0x→ [ ] ( ) ( )∞−==→ 00ln0xlnxlim0x [ ] ∞ ∞− === →→ 0 1 0ln x 1 xln limxlnxlim :functionequivalentantofunctiongiventhegminTransfor 0x0x [ ] 0xlnxlim 0x =∴ → [ ] ( ) ( ) 0xlim x 1 1 x 1 lim x 1 dx d xln dx d lim x 1 xln lim :LHRApply 0x 2 0x0x0x =−= − = = →→→→
- 19. . − −→ 1x 1 xln 1 lim.3 1x ∞−∞=−= − −= − −→ 0 1 0 1 11 1 1ln 1 1x 1 xln 1 lim1x ( ) ( ) ( ) ( ) 0 0 1ln1-1 1ln11 xln1-x xln1x lim 1x 1 xln 1 lim :fractionsimpleaotgminTransfor 1x1x = −− = −− = − − →→ ( )[ ] ( )[ ] ( ) ( ) ( ) ( )( )1xln1 x 1 1x 1 x 1 1 lim xln1-x dx d xln1x dx d lim 1x 1 xln 1 lim :LHRApply 1x1x1x + − − = −− = − − →→→ ( ) ( ) 0 0 1ln111 11 xlnx1-x 1x lim x xlnx1-x x 1-x lim 1x1x = +− − = + − = + ⇒ →→
- 20. . ( ) [ ] [ ] ( ) ( )( )1xln1 x 1 x1 1 lim xlnx1-x dx d 1-x dx d lim xlnx1-x 1-x lim :LHRagainApply 1x1x1x ++ = + = + ⇒ →→→ ( ) 2 1 1ln2 1 xln2 1 lim1x = + = + ⇒ → 2 1 1x 1 xln 1 lim 1x = − −∴ →
- 21. . −→ x2secx 1 x 1 lim.4 220x ( ) ∞∞=−=−= −→ - 0 1 0 1 0sec0 1 0 1 x2secx 1 x 1 lim 220x ( ) 0 0 0 02cos1 x cos2x-1 lim x x2cos x 1 lim x2secx 1 x 1 lim :functionequivalentthetogminTransfor 20x220x220x = − = = −= − →→→ [ ] [ ] ( )( ) ( ) 0 0 0 02sin x2 2x2sin lim x dx d cos2x-1 dx d lim x cos2x-1 lim :LHRApply 0x 2 0x20x == −− == →→→ ( ) ( ) ( ) ( )( )( ) 20cos2x2cos2lim 1 12x2cos lim x dx d x2sin dx d lim x x2sin lim :againLHRApply 0x0x0x0x ===== →→→→ 2 x2secx 1 x 1 lim 220x = −∴ →
- 22. . [ ]x 0x x2lim.5 → [ ] ( )[ ] 00x 0x 002x2lim ==→ [ ]x x2yLet = [ ] [ ] x 1 x2ln 2xlnxyln x2lnyln x == = ( ) ∞ ∞− = ∞ === →→ 0ln 0 1 02ln x 1 x2ln limylnlim :sidesbothonitlimtheApply 0x0x [ ] ( ) ( ) 0xlim x 1 2 x2 1 x 1 dx d x2ln dx d lim x 1 x2ln lim :LHRApply 0x 2 0x0x =−= − = = →→→ ( ) ( ) 12xlimthereforethen 2xysince 1ylimeylim :sidesbothoffunctioninversetheTake x 0x x 0x 0 0x = = =→= → →→ 0ylnlim 0 x 1 x2ln limylnlim 0x 0x0x = == → →→
- 23. . ( ) 1x 1 1x xlim.6 − → + ( ) ( ) ( ) ( )∞ −− → ===+ 111xlim 0 1 11 1 1x 1 1x ( ) 1x 1 xyLet −= ( ) ( ) 1x xln xln 1x 1 xlnyln 1x 1 − = − == − 0 0 11 1ln 1x xln limylnlim :1xassidesbothonitlimtheApplying 1x1x = − = − = → ++ →→ + ( ) ( ) ( ) 1 1 x 1 lim 1 1 x 1 lim 1x dx d xln dx d lim 1x xln lim :memberrighttheonLHRApply 1x1x1x1x === − = − ++++ →→→→
- 24. . ( ) ( ) 72.2exlim xybuteylim :sidesbothoffunctioninversethetake,1ylnlim 1x xln lim ,Thus 1x 1 1x 1x 1 1 1x 1x1x ==∴ == == − − → − → →→ + + ++ ( )x 0x xcotlim.7 + → ( ) ( ) 00x 0x 0cotxcotlim ∞==+ → ( ) ( ) x 1 xcotln xcotlnxxcotlnyln xcotyLet x x === =
- 25. ( ) ( ) ∞ ∞ = ∞ ∞ = = + ++ → →→ ln 0 1 0cotln lim x 1 xcotln limylnlim :sidesbothonlimitthepplyA 0x 0x0x ( ) ( )( ) 2 2 0x0x0x x 1 1xcsc xcot 1 lim x 1 dx d xcotln dx d lim x 1 xcotln lim :memberrightonLHRApply − − = = +++ →→→ xcosxsin2 x2 lim x 1 xcosxsin 1 lim x 1 xsin 1 xcos xsin lim 2 0x 2 0x 2 2 0x ⋅ ⋅ == = +++ →→→ ( ) ( ) 0 0 0sin 02 x2sin x2 lim 22 0x === + →
- 26. ( ) ( ) ( )( ) x2cos x2 lim 2x2cos x4 lim x2sin dx d x2 dx d lim x2sin x2 lim :againLHRApply 0x0x 2 0x 2 0x ++++ →→→→ === ( ) ( ) 0 1 0 0cos 02 === ( ) ( ) 1xcotlimthenxcotySince 1eylim sidesbothoffunctioninversethetake,0ylnlim x 1 xcotln lim ,Hence x 0x x 0 0x 0x0x =∴= == == + + ++ → → →→
- 27. x4sin xtanx2 lim.1 0x + → − → 220y y 1 ysin 1 lim.2 xsin x2 lim.3 10x −→ → ycosln y lim.4 2 0y ( ) x3 x2ln lim.5 3 x +∞→ ( ) − + −→ x2tan 1 x1ln 1 lim.8 10x ( )x 4 2 0x x1lim.9 + → +∞→ x2 2 x e x3 lim.10 ( ) 2 x 2 2 0x xsin1lim.11 + → ( )x 2 x 0x x3elim.12 + → x2tanln x2cosln lim.13 4 x π →( )x 1 0x x2sinx2coslim.6 − → ( )( )xcscxsinlim.15 1 0x − → x x 2 0x e1lim.7 ++ → ( ) xlnxcoslim.14 1 0x − → + EXERCISES: Evaluate the following limits.