Limites Trigonometricos - Exercícios - Ciência da Computação, Notas de estudo de Ciência da Computação. Centro Federal de Educação Tecnológico (CEFET-PA)
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Garoto7 de Março de 2013

Limites Trigonometricos - Exercícios - Ciência da Computação, Notas de estudo de Ciência da Computação. Centro Federal de Educação Tecnológico (CEFET-PA)

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Apostilas e exercicios de Ciência da Computação sobre o estudo dos Limites Trigonometricos.
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Microsoft Word - LimTrigo1.doc

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

1

Usar o limite fundamental e alguns artifícios : 1lim 0

= → x

senx x

1. x

x x sen lim

0→ = ? ‡

x x

x sen lim

0→ =

0 0 , é uma indeterminação.

x x

x sen lim

0→ =

x xx sen

1lim 0→

=

x x

x

senlim

1

0→

= 1 logo x

x x sen lim

0→ = 1

2. x

x x

4senlim 0→

= ? ‡ x

x x

4senlim 0→

= 0 0 ‡

x x

x 4 4sen.4lim

0→ = 4.

y y

y

senlim 0→

=4.1= 4 logo

x x

x

4senlim 0→

=4

3. x

x x 2

5senlim 0→

= ? ‡ = → x

x x 5

5sen. 2 5lim

0 =

y y

y

sen .

2 5lim

0 2 5 logo

x x

x 2 5senlim

0→ =

2 5

4. nx

mx x

senlim 0→

= ? ‡ nx

mx x

senlim 0→

= mx

mx n m

x

sen.lim 0→

= n m .

y y

y

senlim 0→

= n m .1=

n m logo

nx mx

x

senlim 0→

= n m

5. x x

x 2sen 3senlim

0→ = ? ‡

x x

x 2sen 3senlim

0→ = =

x x

x x

x 2sen

3sen

lim 0

= →

x x

x x

x

2 2sen.2

3 3sen.3

lim 0

. 2 3

2 2senlim

3 3senlim

0

0 =

x x

x x

x

x . 1. 2 3

senlim

senlim

0

0 =

t t

y y

t

y =

2 3 logo

x x

x 2sen 3senlim

0→ =

2 3

6. sennx senmx

x 0 lim →

= ? ‡ nx mx

x sen senlim

0→ =

x nx

x mx

x sen

sen

lim 0→

=

nx nxn

mx mxm

x sen.

sen. lim

0→ =

nx nx

mx mx

n m

x sen

sen

.lim 0→

= n m Logo

sennx senmx

x 0 lim →

= n m

7. = → x

tgx x 0 lim ? ‡ =

x tgx

x 0 lim

0 0 ‡ =

x tgx

x 0 lim =

x x x

x cos sen

lim 0

= → xx

x x

1. cos senlim

0

xx x

x cos 1.senlim

0→ =

xx x

xx cos 1lim.senlim

00 →→ = 1 Logo =

x tgx

x 0 lim 1

8. ( ) 1 1lim 2

2

1 − −

a atg

a = ? ‡ ( )

1 1lim 2

2

1 − −

a atg

a =

0 0 ‡ Fazendo

  

→ →

−= 0 1

,12 t x

at ‡ ( ) t ttg

t 0 lim →

=1

logo ( ) 1 1lim 2

2

1 − −

a atg

a =1

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

2

9. xx xx

x 2sen 3senlim

0 + −

→ = ? ‡

xx xx

x 2sen 3senlim

0 + −

→ =

0 0 ‡ ( )

xx xxxf

2sen 3sen

+ −

= =   

   +

  

   −

x xx

x xx

5sen1.

3sen1. =

  

   +

  

   −

x xx

x xx

.5 5sen.51.

.3 3sen.31.

=

x x

x x

.5 5sen.51

.3 3sen.31

+

− ‡

0 lim →x

x x

x x

.5 5sen.51

.3 3sen.31

+

− =

51 31

+ − =

6 2− =

3 1

− logo

xx xx

x 2sen 3senlim

0 + −

→ =

3 1

10. 30 senlim

x xtgx

x

− →

= ? ‡ 30 senlim

x xtgx

x

− →

= xx

x xx

x x cos1

1.sen. cos

1.senlim 2

2

0 +→ =

2 1

( ) 3

sen x

xtgx xf

− = = 3

sen cos sen

x

x x x

− =

3 cos

cos.sensen

x x

xxx

= ( ) xx

xx cos.

cos1.sen 3

− = x

x xx

x cos

cos1.1.sen 2 − =

x x

x x

xx x

cos1 cos1.

cos cos1.1.sen

2 + +− =

xx x

xx x

cos1 1.cos1.

cos 1.sen 2

2

+ − =

xx x

xx x

cos1 1.sen.

cos 1.sen

2

2

+

Logo 30 senlim

x xtgx

x

− →

= 2 1

11. 30 sen11

lim x

xtgx x

+−+ →

=? ‡ xtgxx

xtgx x sen11

1.senlim 30 +++ −

→ =

xtgxxx x

xx x

x sen11 1.

cos1 1.sen.

cos 1.senlim 2

2

0 ++++→ =

2 1.

2 1.

1 1.

1 1.1 =

4 1

( ) 3

11

x

senxtgx xf

+−+ = =

xtgxx xtgx

sen11 1.sen11 3 +++

−−+ = xtgxx

xtgx sen11

1.sen3 +++ −

30

sen11 lim

x xtgx

x

+−+ →

= 4 1

12. ax

ax ax

− →

sensenlim = ? ‡ ax

ax ax

− →

sensenlim =   

   −

  

   +

  

   −

2 .2

2 cos.

2 sen2

lim ax

axax

ax =

1 2

cos. .

2 .2

) 2

sen(2 lim

  

   +

  

   −

ax

ax

ax

ax = acos Logo

ax ax

ax − −

sensenlim = cosa

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

3

13. ( ) a

xax a

sensenlim 0

−+ →

= ? ‡ ( ) a

xax a

sensenlim 0

−+ →

= 1

2 cos.

.

2 .2

2 sen2

lim   

   ++

  

   −

  

   −+

xax

ax

xax

aa =

1 2

2cos. .

2 .2

2 sen2

lim   

   +

  

  

  

  

ax

a

a

aa = xcos Logo ( )

a xax

a

sensenlim 0

−+ →

=cosx

14. ( ) a

xax a

coscoslim 0

−+ →

= ? ‡ ( ) a

xax a

coscoslim 0

−+ →

= a

xaxxax

a

  

   −−

  

   ++−

2 sen.

2 sen2

lim 0

=

  

   −

  

   −

  

   +−

2 .2

2 sen.

2 2sen.2

lim 0 a

aax

a =

  

   −

  

   −

  

   +−

2

2 sen

. 2

2senlim 0 a

a ax

a = xsen− Logo

( ) a

xax a

coscoslim 0

−+ →

=-senx

15. ax

ax ax

− →

secseclim = ? ‡ ax

ax ax

− →

secseclim = ax

ax ax

cos 1

cos 1

lim = ax

ax xa

ax

cos.cos coscos

lim =

( ) axax xa

ax cos.cos. coscoslim

− −

→ = ( ) axax

xaxa

ax cos.cos. 2

sen. 2

sen.2 lim

  

   −

  

   +−

→ =

axxa

xaxa

ax cos.cos 1.

2 .2

2 sen

. 1

2 sen.2

lim   

   −−

  

   −

  

   +−

→ =

axxa

xaxa

ax cos.cos 1.

2

2 sen

. 1

2 sen

lim   

   −

  

   −

  

   +

→ =

aa a

cos.cos 1.1.

1 sen =

aa a

cos 1.

cos sen = atga sec. Logo

ax ax

ax − −

secseclim = atga sec.

16. x

x x sec1 lim

2

0 −→ = ? ‡

x x

x sec1 lim

2

0 −→ =

( )xxx xx

cos1 1.

cos 1.sen 1lim

2

20

+ −

→ = 2−

( )

x

xxf

cos 11

2

− = =

x x

x

cos 1cos

2

− = ( )x

xx cos1.1

cos.2

−− = ( ) ( )

( )x x

xx x

cos1 cos1.

cos 1.cos1

1

2 + +−

− =

( )xxx x

cos1 1.

cos 1.cos1

1

2

2

+ −

− =

( )xxx x

cos1 1.

cos 1.sen 1

2

2

+ −

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

4

17. tgx

gx

x − −

→ 1 cot1

lim 4 π

= ? ‡ tgx

gx

x − −

→ 1 cot1lim

4 π

= tgx tgx

x

→ 1

11 lim

4 π

= tgx

tgx tgx

x

→ 1

1

lim 4 π

=

tgx tgx

tgx

x

−−

→ 1

)1.(1

lim 4 π

= tgxx

1lim 4

− →

π = 1− Logo

tgx gx

x − −

→ 1 cot1

lim 4 π

= -1

18. x

x x 2

3

0 sen cos1lim −

→ = ? ‡

x x

x 2

3

0 sen cos1lim −

→ = ( )( )

x xxx

x 2

2

0 cos1 coscos1.cos1lim

− ++−

→ =

( )( ) ( )( )xx

xxx x cos1.cos1

coscos1.cos1lim 2

0 +− ++−

→ =

x xx

x cos1 coscos1lim

2

0 + ++

→ =

2 3 Logo

x x

x 2

3

0 sen cos1lim −

→ =

2 3

19. x

x

x cos.21 3senlim

3 −→π

= ? ‡ x

x

x cos.21 3senlim

3 −→π

= ( ) 1

cos.21.senlim 3

xx

x

+ −

π

= 3−

( ) x

xxf cos.21 3sen

− = = ( )

x xx

cos.21 2sen

− + =

x xxxx

cos.21 cos.2sen2cos.sen

− + = ( )

x xxxxx

cos.21 cos.cos.sen.21cos2.sen 2

− +− =

( )[ ] x

xxx cos.21

cos21cos2.sen 22

− +− = [ ]

x xx

cos.21 1cos4.sen 2

− − = ( )( )

x coxcoxx

cos.21 .21..21.sen

− +−

− = ( ) 1

cos.21.sen xx + −

20. tgx

xx x

− → 1

cossenlim 4

π = ? ‡

tgx xx

x − −

→ 1 cossenlim

4 π

= ( )x x

coslim 4

− →π

= 2 2

( ) tgx

xxxf − −

= 1

cossen =

x x

xx

cos sen1

cossen

− =

x x

xx

cos sen1

cossen

− =

x xx xx

cos sencos cossen

− − = ( )

x xx

xx

cos cossen.1

cossen −−

− =

xx xxx sencos

cos. 1

cossen −

− − = xcos−

21. ( ) )sec(cos.3lim 3

xx x

π− →

= ? ‡ ( ) )sec(cos.3lim 3

xx x

π− →

= ∞.0

( ) ( ) )sec(cos.3 xxxf π−= = ( ) ( )xx πsen 1.3 − = ( )x

x ππ

− sen

3 = ( )x x

ππ − −

3sen 3 = ( )

( )x x

− −

3. 3sen. 1

π πππ

=

( ) ( )x

x ππ

πππ

− 3

3sen. 1 ‡ ( ) )sec(cos.3lim

3 xx

x π

→ = ( )

( )x xx

ππ πππ

− −→

3 3sen. 1lim

3 =

π 1

22. )1sen(.lim x

x x→∝

= ? ‡ )1sen(.lim x

x x→∝

= 0.∞

x

x x 1

1sen lim

  

  

→∝ = 1senlim

0 =

t t

t ‡ Fazendo

  

→ +∞→

= 0

1 t x

x t

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

5

23. 1sen.3sen.2

1sensen.2lim 2 2

6 +− −+

xx xx

x π = ? ‡

1sen.3sen.2 1sensen.2lim 2

2

6 +−

−+ → xx

xx x π

= x

x x sen1

sen1lim 6 +−

+ →π

=

6 sen1

6 sen1

π

π

+−

+ =

2 11

2 11

+−

+ = 3− ‡ ( )

1sen.3sen.2 1sensen.2

2

2

+− −+

= xx

xxxf = ( )

( )1sen. 2 1sen

1sen. 2 1sen

−  

   −

+  

   −

xx

xx = ( )( )1sen

1sen − +

x x =

x x

sen1 sen1

+− +

24. ( )   

  −

→ 2 .1lim

1

xtgx x

π = ? ‡ ( )   

  −

→ 2 .1lim

1

xtgx x

π = ∞.0 ‡ ( ) ( )   

  −=

2 .1 xtgxxf π =

( )   

   −−

22 cot.1 xgx ππ = ( )

  

   −

22

1 xtg

x ππ

= ( )

  

   −

22

2.1. 2

xtg

x

ππ π

π

=

( )x

xtg

  

   −

1. 2

22

2

π

ππ π =

  

   −

  

   −

22

22

2

x

xtg

ππ

ππ π

( )   

  −

→ 2 .1lim

1

xtgx x

π =

  

   −

  

   −

22

22

2

lim 1

x

xtg x

ππ

ππ π = ( )

t ttg

t 0 lim

2

π = π 2 Fazendo uma mudança de variável,

temos :   

→ →

−= 0 1

2 t x

x xt ππ

25. ( )x x

x πsen 1lim

2

1

− →

= ? ‡ ( )x x

x πsen 1lim

2

1

− →

= ( ) ( )x

x x

x

ππ πππ

− −

+ → sen.

1lim 1

= π 2

( ) x

xxf πsen

1 2− = = ( )( )( )x

xx ππ − +−

sen 1.1 = ( )

( )x x

x

− −

+

1 sen

1 ππ

= ( ) ( )x

x x

− −

+

1. sen.

1

π πππ

= ( ) ( )x

x x

ππ πππ

− −

+ sen.

1

26.   

   −

xgxg

x 2 cot.2cotlim

0

π = ? ‡   

   −

xgxg

x 2 cot.2cotlim

0

π = 0.∞

( )   

   −= xgxgxf

2 cot.2cot π = tgxxg .2cot =

xtg tgx

2 =

xtg tgx

tgx

21 2 −

= tgx

xtgtgx .2

1. 2− =

2 1 2 xtg

  

   −

xgxg

x 2 cot.2cotlim

0

π = 2

1lim 2

0

xtg x

− →

= 2 1

27. x

xx x 2

3

0 sen coscoslim −

→ = 11102

2

1 ...1 lim

tttt t

t +++++ −

→ =

12 1

( ) x

xxxf 2 3

sen coscos −

= = 12 23

1 t tt

− = ( ) ( )( )11102

2

...1.1 1.

ttttt tt

+++++− −− = 11102

2

...1 tttt t

+++++ −

63.2 coscos xxt ==   

→ →

1 0

t x xt cos6 = , xt 212 cos= , 122 1sen tx −=

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

6

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

28. xx xx

x sencos 12cos2senlim

4 − −−

π = ? ‡

xx xx

x sencos 12cos2senlim

4 − −−

π = ( )x

x cos.2lim

4

− →π

= 4

cos.2 π− = 2 2.2− =

2−

( ) xx xxxf

sencos 12cos2sen

− −−

= = ( ) xx

xxx sencos

11cos2cossen.2 2

− −−− =

xx xxx

sencos 11cos2cos.sen.2 2

− −+− =

xx xxx

sencos cos2cos.sen.2 2

− − = ( )

xx xxx

sencos sencos.cos.2

− −− = xcos.2−

29. ( ) 112

1senlim 1 −−

− → x

x x

= ? ‡ ( ) 112

1senlim 1 −−

− → x

x x

= ( )( ) 1 112.

1 1sen.

2 1lim

1

+− −

− →

x x

x x

= 1

( ) ( ) 112

1sen −−

− =

x xxf = ( )

112 112.

112 1sen

+−

+−

−−

x x

x x = ( )

1 112.

112 1sen +−

−− − x

x x = ( )( ) 1

112. 1.2 1sen +−

− − x

x x =

( ) ( ) 1

112. 1

1sen. 2 1 +−

− − x

x x

30.

3

cos.21lim 3

ππ

x

x

x = ? ‡

3

cos.21lim 3

ππ

x

x

x =

 

 

 −

 

 

 −

 

 

 +

2 3

2 3sen

. 2

3sen.2lim 3 x

x

x

x π

π

π

π =

. 2

33sen.2  

 

 + ππ = .

2 3

2 sen.2

 

 

π = .

3 sen.2 

 

   π = 3

2 3.2 =

( )

3

cos.21 π

− =

x

xxf =

3

cos 2 1.2

π

  

   −

x

x =

3

cos 3

cos.2

π

π

  

   −

x

x =

( )

 

 

 − −

 

 

 −

 

 

 + −

2 3.2.1

2 3sen.

2 3sen2.2

x

xx

π

ππ

=

 

 

 −

 

 

 −

 

 

 +

2 3

2 3sen.

2 3sen.2

x

xx

π

ππ

=

 

 

 −

 

 

 −

 

 

 +

2 3

2 3sen

. 2

3sen.2 x

x

x

π

π

π

31. xx x

x sen. 2cos1lim

0

− →

= ? ‡ xx x

x sen. 2cos1lim

0

− →

= x

x

x

sen.2lim 3 π

→ = 2

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

7

( ) xx xxf

sen. 2cos1 −

= = ( ) xx

x sen.

sen211 2−− = xx

x sen.

sen211 2+− = xx x

sen. sen.2 2 =

x xsen.2

32. xx

x x sen1sen1 lim

0 −−+→ = ? ‡

xx x

x sen1sen1 lim

0 −−+→ =

x x

xx x sen.2

sen1sen1lim 0

−++ →

= 1.2 11+

=1

( ) xx

xxf sen1sen1 −−+

= = ( )( )xx xxx

sen1sen1 sen1sen1.

−−+ −++ = ( )

xx xxx

sen1sen1 sen1sen1.

+−+ −++ =

( ) x

xxx sen.2

sen1sen1. −++ =

x x

xx sen.2

sen1sen1 −++ = 1.2 11+ = 1

33. xx

x x sencos

2coslim 0 −→

= 1

sencoslim 0

xx x

+ →

= 2 2

2 2

+ = 2

( ) xx

xxf sencos 2cos

− = = ( )( )( )xxxx

xxx sencos.sencos

sencos.2cos +−

+ = ( ) xx

xxx 22 sencos sencos.2cos

+ = ( ) x

xxx 2cos

sencos.2cos + =

( ) x

xxx 2cos

sencos.2cos + = 1

sencos xx + = 2 2

2 2

+ = 2

34.

3

sen.23lim 3

ππ

x

x

x = ? ‡

3

sen.23lim 3

ππ

x

x

x =

3

sen 2 3.2

lim 3

ππ

  

   

 −

x

x

x =

3

sen 3

sen.2 lim

3 π

π

π

  

   −

x

x

x =

3

2 3cos.

2 3sen.2

lim 3

π

ππ

π

   

   

   

   

 +

   

   

 −

x

xx

x =

3 3

2 3

3

cos. 2 3

3

sen.2

lim 3

π

ππ

π

   

   

   

   

 +

   

   

 −

x

xx

x =

( ) 3

3.1 6

3cos. 6 3sen.2

lim 3

x

xx

x −−

 

  

   

   +

  

   −

π

ππ

π

35. ?

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