# Exercicios Resolvidos de Limites Trigonometricos

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

 Usar o limite fundamental e alguns artifícios : 1lim0=

1 fi xsenxx

1. x x senlim

 = ? à x

0fi x senlim

0fi= 0

0 , é uma indeterminação.

x x senlim

0fi x sen 1lim 0fi = x x senlim

0fi

= 1 logo x x senlim

0fi

2. x

4senlim

0fi = ? à x

4senlim

 =4.1= 4 logo

senlim 0fi x x

4senlim 0fi =4

 = ? à =

5senlim 0fi sen

4. nxmxx senlim

0fi = ? à nxmxx senlim 0fi = mxmxnmx sen.lim senlim

0fi = n m .1= n m logo nxmxx senlim 0fi = n m

5. x x 2sen

3senlim

0fi

= ? à x x 2sen

3senlim 0fi = = fi x x x x x

2 2senlim

3 3senlim x x senlim fifi t t y t y=

2 3 logo x x 2sen

6. sennxsenmx x 0limfi

= ? à nxmx x sen senlim 0fi = nx x x sen sen lim 0fi = nx nxn mx mxm sen.

lim 0fi = nx mx nm x sen sen

.lim 0fi = n m Logo sennxsenmx x 0limfi = n fi xtgx x 0

 lim ? à =

fi xtgx x 0lim 0 fi xtgx x 0 fi x x cossen fi x 1.cos senlim0 x x x cos

1.senlim

0fi = x x x cos fi xtgx x 0 lim1

atga= 0

0 à Fazendo î íì fi

 at à ()t

ttg atga

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

9. x x x x x x x x x x x à 0lim fix x x x x x

 = ? à 30

xtgxx -fi senlim x xtgxx-fi= x x

1.senlim xtgx sen x x cos.sensen x x x x x cos1 cos1.cos x x x x

senlim x xtgxx -fi = 2 sen11 lim xtgxx xtgx xtgxxx x x x x sen11 senxtgx xtgxx xtgx xtgx sen11 lim

12. ax ax --fi sensenlim= ? à ax ax --fi sensenlim= sen2 lim ax axax ax = ax = acos Logo ax --fi sensenlim= cosa

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos xaxa sensenlim0 -+fi xaxa sensenlim0 xax xax æfi axaa a = xcos Logo ()a xaxa sensenlim0 -+fi

=cosx xaxa coscoslim0 -+fi xaxa coscoslim0 -+fi= xaxxax sen2 lim0 = aax

2 sen axa = xsen- Logo xaxa coscoslim0 -+fi

=-senx

15. ax ax --fi secseclim= ? à ax ax --fi secseclim= ax cos1cos1 lim= cos.cos coscos lim=

() axax xa ax cos.cos. coscoslim - -fi = () axax xaxa lim - fi = axxa xaxa lim fi= axxa xaxa

2 sen

2 sen lim fi = sen= a cos 1.cos sen= atgasec. Logo ax ax --fi secseclim=atgasec.

16. x x sec1 lim lim

0 -fi =

() x

1lim

() x xxf cos x x cos

() x x x cos1 cos1.cos

() x x

() x x

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

17. tgxgx cot1 lim p = ? à tgxgx cot1lim p= tgxtgx

4 p =tgx tgx tgx lim p = tgx tgx tgx fi 1 lim p = tgxx p =1- Logo tgxgx cot1 lim p = -1 x cos1 coscos1lim

3 Logo x

3senlim

-fi p = ? à x

3senlim

-fi fi p

3sen coxcoxx

20. tgx x cossenlim p = ? à tgx cossenlim

() tgx xxxf -

1 cossen = x x cos sen1 cossen- x x cos sen1 cossen- x x cos sencos cossen - x x cos cossen.1 cossen x x sencos cos.1 cossen - p =

 1 à ())sec(cos.3lim3

() x x xfiµ = ? à )1sen(.limxx xfiµ fi t t à Fazendo î íì fi

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos x p = ? à x p = x x sen1 sen1lim

 =3- à ()

x xxxf = sen1 fi 2

 p= ? à ()÷

fi 2

.1xtgxxfp= xtg x p xtg x p p = p p = x xtg p

 p à

fi 2

2 lim xtg x ttg

 2 Fazendo uma mudança de variável, temos : î

p =p íì fi

xtpp

25. ()x x psen

1lim

= ? à ()x x psen x x p p - x x x x p p= () x x p p -

Ł æ -fi xgxg p= ? à ÷

Ł æ -fi xgxg

Ł æ -= xgxgxf 2 xtg tgx tgx

= tgx xtgtgx

Ł æ -fi xgxg

1lim

 1 1lim

1. t t

 63.2coscosxxt== î

íì fifi10t

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

 BriotxRuffini : 1 0 0 0 -1 1 • 1 1 1 1 1 1 1 1 0

28. x x sencos 12cos2senlim fip = ? à x x sencos 12cos2senlim fip = ()x

() x xxxf sencos 12cos2sen - x x sencos sencos.cos.2 - x=

 = ? à 3

x x p p = æ +p æ p x xxf = æ -x x = x xxp p = x xxp x p p p x x x sen.

 = ? à

x x x sen.

sen.2lim p fi

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos x x sen.

x x sen.

x x sen.

x x x x x x x sencos

2coslim

() x xxf sencos

2cos x sencos.sencos sencos.2cos

2 sencos sencos.2cos -

2cos sencos.2cos+ =

 = ? à 3

lim fi x x x lim p p fi x x x lim fi x x = lim fi x x = lim fi p p p