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# Limites trigonometricos

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### Limites trigonometricos

1. 1. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos1Usar o limite fundamental e alguns artifícios : 1lim0=→ xsenxx1.xxx senlim0→= ? àxxx senlim0→=00, é uma indeterminação.xxx senlim0→=xxx sen1lim0→=xxxsenlim10→= 1 logoxxx senlim0→= 12.xxx4senlim0→= ? àxxx4senlim0→=00àxxx 44sen.4lim0→= 4.yyysenlim0→=4.1= 4 logoxxx4senlim0→=43.xxx 25senlim0→= ? à =→ xxx 55sen.25lim0=→ yyysen.25lim0 25logoxxx 25senlim0→=254.nxmxxsenlim0→= ? ànxmxxsenlim0→=mxmxnmxsen.lim0→=nm.yyysenlim0→=nm.1=nmlogonxmxxsenlim0→=nm5.xxx 2sen3senlim0→= ? àxxx 2sen3senlim0→= =→xxxxx 2sen3senlim0=→xxxxx22sen.233sen.3lim0.2322senlim33senlim00=→→xxxxxx. 1.23senlimsenlim00=→→ttyyty=23logoxxx 2sen3senlim0→=236.sennxsenmxx 0lim→= ? ànxmxx sensenlim0→=xnxxmxx sensenlim0→=nxnxnmxmxmx sen.sen.lim0→=nxnxmxmxnmx sensen.lim0→=nmLogosennxsenmxx 0lim→=nm7. =→ xtgxx 0lim ? à =→ xtgxx 0lim00à =→ xtgxx 0lim =→ xxxxcossenlim0=→ xxxx1.cossenlim0xxxx cos1.senlim0→=xxxxx cos1lim.senlim00 →→= 1 Logo =→ xtgxx 0lim 18.( )11lim 221 −−→ aatga= ? à( )11lim 221 −−→ aatga=00àFazendo→→−=01,12txat à( )tttgt 0lim→=1logo( )11lim 221 −−→ aatga=1
2. 2. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos29.xxxxx 2sen3senlim0 +−→= ? àxxxxx 2sen3senlim0 +−→=00à ( )xxxxxf2sen3sen+−= =+−xxxxxx5sen1.3sen1.=+−xxxxxx.55sen.51..33sen.31.=xxxx.55sen.51.33sen.31+−à0lim→xxxxx.55sen.51.33sen.31+−=5131+−=62−=31− logoxxxxx 2sen3senlim0 +−→=31−10. 30senlimxxtgxx−→= ? à 30senlimxxtgxx−→=xxxxxxx cos11.sen.cos1.senlim 220 +→=21( ) 3senxxtgxxf−= = 3sencossenxxxx−= 3coscos.sensenxxxxx −=( )xxxxcos.cos1.sen3−=xxxxxcoscos1.1.sen2−=xxxxxxxcos1cos1.coscos1.1.sen2 ++−=xxxxxxcos11.cos1.cos1.sen22+−=xxxxxxcos11.sen.cos1.sen22+Logo 30senlimxxtgxx−→=2111. 30sen11limxxtgxx+−+→=? àxtgxxxtgxx sen111.senlim 30 +++−→=xtgxxxxxxxx sen111.cos11.sen.cos1.senlim 220 ++++→=21.21.11.11.1 =41( ) 311xsenxtgxxf+−+= =xtgxxxtgxsen111.sen113+++−−+=xtgxxxtgxsen111.sen3+++−30sen11limxxtgxx+−+→=4112.axaxax −−→sensenlim = ? àaxaxax −−→sensenlim = − + −→2.22cos.2sen2limaxaxaxax=12cos..2.2)2sen(2lim + −−→axaxaxax= acos Logoaxaxax −−→sensenlim = cosa
3. 3. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos313.( )axaxasensenlim0−+→= ? à( )axaxasensenlim0−+→=12cos..2.22sen2lim ++ − −+→xaxaxxaxaa=122cos..2.22sen2lim +→axaaaa= xcos Logo( )axaxasensenlim0−+→=cosx14.( )axaxacoscoslim0−+→= ? à( )axaxacoscoslim0−+→=axaxxaxa −− ++−→2sen.2sen2lim0= − − +−→2.22sen.22sen.2lim0 aaaxa= − − +−→22sen.22senlim0 aaaxa= xsen− Logo( )axaxacoscoslim0−+→=-senx15.axaxax −−→secseclim = ? àaxaxax −−→secseclim =axaxax −−→cos1cos1lim =axaxxaax −−→cos.coscoscoslim =( ) axaxxaax cos.cos.coscoslim−−→=( ) axaxxaxaax cos.cos.2sen.2sen.2lim− − +−→=axxaxaxaax cos.cos1.2.22sen.12sen.2lim −− − +−→=axxaxaxaax cos.cos1.22sen.12senlim − − +→=aaacos.cos1.1.1sen=aaacos1.cossen= atga sec. Logoaxaxax −−→secseclim = atga sec.16.xxx sec1lim20 −→= ? àxxx sec1lim20 −→=( )xxxxxcos11.cos1.sen1lim220+−→= 2−( )xxxfcos112−= =xxxcos1cos2−=( )xxxcos1.1cos.2−−=( ) ( )( )xxxxxcos1cos1.cos1.cos112 ++−−=( )xxxxcos11.cos1.cos1122+−−=( )xxxxcos11.cos1.sen122+−
4. 4. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos417.tgxgxx −−→ 1cot1lim4π= ? àtgxgxx −−→ 1cot1lim4π=tgxtgxx −−→ 111lim4π=tgxtgxtgxx −−→ 11lim4π=tgxtgxtgxx −−−→ 1)1.(1lim4π=tgxx1lim4−→π= 1− Logotgxgxx −−→ 1cot1lim4π= -118.xxx 230 sencos1lim−→= ? àxxx 230 sencos1lim−→=( )( )xxxxx 220 cos1coscos1.cos1lim−++−→=( )( )( )( )xxxxxx cos1.cos1coscos1.cos1lim20 +−++−→=xxxx cos1coscos1lim20 +++→=23Logoxxx 230 sencos1lim−→=2319.xxx cos.213senlim3−→π= ? àxxx cos.213senlim3−→π=( )1cos.21.senlim3xxx+−→π= 3−( )xxxfcos.213sen−= =( )xxxcos.212sen−+=xxxxxcos.21cos.2sen2cos.sen−+=( )xxxxxxcos.21cos.cos.sen.21cos2.sen 2−+−=( )[ ]xxxxcos.21cos21cos2.sen 22−+−=[ ]xxxcos.211cos4.sen 2−−=( )( )xcoxcoxxcos.21.21..21.sen−+−− =( )1cos.21.sen xx +−20.tgxxxx −−→ 1cossenlim4π= ? àtgxxxx −−→ 1cossenlim4π= ( )xxcoslim4−→π=22−( )tgxxxxf−−=1cossen=xxxxcossen1cossen−−=xxxxcossen1cossen−−=xxxxxcossencoscossen−−=( )xxxxxcoscossen.1cossen−−−=xxxxxsencoscos.1cossen−−− = xcos−21. ( ) )sec(cos.3lim3xxxπ−→= ? à ( ) )sec(cos.3lim3xxxπ−→= ∞.0( ) ( ) )sec(cos.3 xxxf π−= =( )( )xxπsen1.3 − =( )xxππ −−sen3=( )xxππ −−3sen3=( )( )xx−−3.3sen.1ππππ=( )( )xxπππππ−−33sen.1à ( ) )sec(cos.3lim3xxxπ−→=( )( )xxxπππππ−−→33sen.1lim3=π122. )1sen(.limxxx→∝= ? à )1sen(.limxxx→∝= 0.∞xxx 11senlim→∝= 1senlim0=→ tttà Fazendo→+∞→=01txxt
5. 5. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos523.1sen.3sen.21sensen.2lim 226 +−−+→ xxxxx π= ? à1sen.3sen.21sensen.2lim 226 +−−+→ xxxxx π=xxx sen1sen1lim6+−+→π=6sen16sen1ππ+−+=211211+−+= 3− à ( )1sen.3sen.21sensen.222+−−+=xxxxxf =( )( )1sen.21sen1sen.21sen−−+−xxxx=( )( )1sen1sen−+xx=xxsen1sen1+−+24. ( ) −→ 2.1lim1xtgxxπ= ? à ( ) −→ 2.1lim1xtgxxπ= ∞.0 à ( ) ( ) −=2.1xtgxxfπ=( ) −−22cot.1xgxππ=( )−−221xtgxππ=( )−−222.1.2xtgxππππ=( )xxtg−−1.2222ππππ =−−22222xxtgπππππ à( ) −→ 2.1lim1xtgxxπ=−−→22222lim1xxtgxπππππ =( )tttgt 0lim2→π =π2Fazendo uma mudança de variável,temos :→→−=012 txxxtππ25.( )xxx πsen1lim21−→= ? à( )xxx πsen1lim21−→=( )( )xxxxπππππ−−+→ sen.1lim1=π2( )xxxfπsen1 2−= =( )( )( )xxxππ −+−sen1.1=( )( )xxx−−+1sen1ππ=( )( )xxx−−+1.sen.1ππππ=( )( )xxxπππππ−−+sen.126. −→xgxgx 2cot.2cotlim0π= ? à −→xgxgx 2cot.2cotlim0π= 0.∞( ) −= xgxgxf2cot.2cotπ= tgxxg .2cot =xtgtgx2=xtgtgxtgx212−=tgxxtgtgx.21.2−=21 2xtg−−→xgxgx 2cot.2cotlim0π=21lim20xtgx−→=2127.xxxx 230 sencoscoslim−→= 1110221 ...1limtttttt +++++−→=121−( )xxxxf 23sencoscos −= = 12231 ttt−−=( )( )( )111022...1.11.ttttttt+++++−−−= 111022...1 ttttt+++++−63.2coscos xxt ==→→10txxt cos6= , xt 212cos= , 1221sen tx −=
6. 6. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos6BriotxRuffini :1 0 0 ... 0 -11 • 1 1 ... 1 11 1 1 ... 1 028.xxxxx sencos12cos2senlim4−−−→π= ? àxxxxx sencos12cos2senlim4−−−→π= ( )xxcos.2lim4−→π=4cos.2π− =22.2− =2−( )xxxxxfsencos12cos2sen−−−= =( )xxxxxsencos11cos2cossen.2 2−−−−=xxxxxsencos11cos2cos.sen.2 2−−+−=xxxxxsencoscos2cos.sen.2 2−−=( )xxxxxsencossencos.cos.2−−−= xcos.2−29.( )1121senlim1−−−→xxx= ? à( )1121senlim1 −−−→ xxx=( )( ) 1112.11sen.21lim1+−−−→xxxx= 1( ) ( )1121sen−−−=xxxf =( )112112.1121sen+−+−−−−xxxx=( )1112.1121sen +−−−− xxx=( )( ) 1112.1.21sen +−−− xxx=( )( ) 1112.11sen.21 +−−− xxx30.3cos.21lim3ππ−−→ xxx= ? à3cos.21lim3ππ−−→ xxx= − − +→2323sen.23sen.2lim3 xxxx ππππ=.233sen.2 +ππ= .232sen.2 π= .3sen.2 π= 323.2 =( )3cos.21π−−=xxxf =3cos21.2π−−xx=3cos3cos.2ππ−−xx=( ) −− − +−23.2.123sen.23sen2.2xxxπππ= − − +2323sen.23sen.2xxxπππ= − − +2323sen.23sen.2xxxπππ31.xxxx sen.2cos1lim0−→= ? àxxxx sen.2cos1lim0−→=xxxsen.2lim3π→= 2
7. 7. Limites Trigonométricos ResolvidosSete páginas e 34 limites resolvidos7( )xxxxfsen.2cos1−= =( )xxxsen.sen211 2−−=xxxsen.sen211 2+−=xxxsen.sen.2 2=xxsen.232.xxxx sen1sen1lim0 −−+→= ? àxxxx sen1sen1lim0 −−+→=xxxxx sen.2sen1sen1lim0−++→=1.211+=1( )xxxxfsen1sen1 −−+= =( )( )xxxxxsen1sen1sen1sen1.−−+−++=( )xxxxxsen1sen1sen1sen1.+−+−++=( )xxxxsen.2sen1sen1. −++=xxxxsen.2sen1sen1 −++=1.211+= 133.xxxx sencos2coslim0 −→=1sencoslim0xxx+→=2222+ = 2( )xxxxfsencos2cos−= =( )( )( )xxxxxxxsencos.sencossencos.2cos+−+=( )xxxxx22sencossencos.2cos−+=( )xxxx2cossencos.2cos +=( )xxxx2cossencos.2cos +=1sencos xx +=2222+ = 234.3sen.23lim3ππ−−→ xxx= ? à3sen.23lim3ππ−−→ xxx=3sen23.2lim3ππ−−→ xxx=3sen3sen.2lim3πππ−−→ xxx=323cos.23sen.2lim3ππππ−+−→ xxxx=33233cos.233sen.2lim3ππππ − + −→xxxx=( )33.163cos.63sen.2lim3xxxx−− + −→ππππ35. ?