Limites Trigonométricos Resolvidos. Sete páginas e 34 limites resolvidos. sen x. = 1 logo = 1. lim = 4. lim. = n. lim. lim lim =? m m m.

(1)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

Usar o limite fundamental e alguns artifícios :

lim 1

0 = → x senx x

1.

x x x sen lim 0 →

= ? à

x x x sen lim 0 →

=

₀

0 , é uma indeterminação.

x x x _sen lim 0 →

=

x x x sen 1 lim 0 →

=

x x x sen lim 1 0 →

= 1 logo

x x x sen lim 0 →

= 1

2.

x x x 4 sen lim 0 →

= ? à

=

0 0

à

x x x 4 4 sen . 4 lim 0 →

= 4.

y y y sen lim 0 →

=4.1= 4 logo

=4

3.

x x x 2 5 sen lim 0 →

= ? à

→ x = x x 5 5 sen . 2 5 lim 0 → y = y y sen . 2 5 lim 0 2 5

logo

x x x 2 5 sen lim 0 →

=

2 5

4.

nx mx x sen lim 0 →

= ? à

=

mx mx n m x sen . lim 0 →

=

n m

.

y y y sen lim 0 →

=

n m

.1=

n m

logo

=

n m

5.

x x x sen2 3 sen lim 0 →

= ? à

=

→ = x x x x x sen2 3 sen lim 0 → = x x x x x 2 2 sen . 2 3 3 sen . 3 lim 0

.

2 3 2 2 sen lim 3 3 sen lim 0 0 ₌ → → x x x x x x

.

.1 2 3 sen lim sen lim 0 0 = → → t t y y t y

=

2 3

logo

=

2 3

6.

sennx senmx x 0 lim →

= ? à

nx mx x sen sen lim 0 →

=

x nx x mx x sen sen lim 0 →

=

nx nx n mx mx m 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

? à

= → x tgx x 0 lim 0 0

à

= → x tgx x 0 lim = → x x x x cos sen lim 0 → x x = x x 1 . cos sen lim 0 x x x x cos 1 . sen lim 0 →

=

x x x x x cos 1 lim . sen lim 0 0 → →

= 1

Logo

→ x = tgx x 0 lim

1

8. ( )

1 1 lim 2 2 1 − − → a a tg a

= ? à

( )

1 1 lim 2 2 1 − − → a a tg a

=

0 0

à Fazendo

   → → − = 0 1 , 1 2 t x a t

à

( )

t t tg t 0 lim →

=1

logo

( )

1 1 lim 2 2 1 − − → a a tg a

=1

(2)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

9.

x x x x x sen2 3 sen lim 0 + − →

= ? à

=

0 0

à

( )

x x x x x f 2 sen 3 sen + − =

=

      +       − x x x x x x 5 sen 1 . 3 sen 1 .

=

      +       − x x x x x x . 5 5 sen . 5 1 . . 3 3 sen . 3 1 .

=

x x x x . 5 5 sen . 5 1 . 3 3 sen . 3 1 + −

à

0 lim → x x x x x . 5 5 sen . 5 1 . 3 3 sen . 3 1 + −

=

5 1 3 1 + −

=

6 2 −

=

3 1 −

logo

=

3 1 −

10.

₃ 0 sen lim x x tgx x − →

= ? à

₀ 3 sen lim x x tgx x − →

=

x x x x x x x 1 cos 1 . sen . cos 1 . sen lim 2 2 0 + →

=

2 1

( )

₃sen x x tgx x f = −

=

₃ sen cos sen x x x x −

=

cos₃ cos . sen sen x x x x x−

=

(

)

x x x x cos . cos 1 . sen 3 −

=

x x x x x cos cos 1 . 1 . sen 2 −

=

x x x x x x x cos 1 cos 1 . cos cos 1 . 1 . sen 2 + + −

=

x x x x x x cos 1 1 . cos 1 . cos 1 . sen 2 2 + −

=

x x x x x x cos 1 1 . sen . cos 1 . sen 2 2 +

Logo

₃ 0 sen lim x x tgx x − →

=

2 1

11.

₃ 0 sen 1 1 lim x x tgx x + − + →

=? à

_x _tgx _x x tgx x ₁ ₁ _sen 1 . sen lim 3 0 + + + − →

=

x tgx x x x x x x x ₁ ₁ _sen 1 . cos 1 1 . sen . cos 1 . sen lim 2 2 0 + + + + →

=

2 1 . 2 1 . 1 1 . 1 1 . 1

=

4 1

( )

1 ₃1 x senx tgx x f = + − +

=

x tgx x x tgx sen 1 1 1 . sen 1 1 3 ₊ ₊ ₊ − − +

=

x tgx x x tgx sen 1 1 1 . sen 3 ₊ ₊ ₊ − 3 0 sen 1 1 lim x x tgx x + − + →

=

4 1

12.

a x a x a x − − → sen sen lim

= ? à

=

      −       +       − → 2 . 2 2 cos . 2 sen 2 lim a x a x a x a x

=

1 2 cos . . 2 . 2 ) 2 sen( 2 lim       +       − − → a x a x a x a x

=

a cos

Logo

= cosa

(3)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

13. (

)

a x a x a sen sen lim 0 − + →

= ? à

(

)

=

1 2 cos . . 2 . 2 2 sen 2 lim       + +       −       + − → x a x a x x a x a a

=

1 2 2 cos . . 2 . 2 2 sen 2 lim       +             → a x a a a a

=

x cos

Logo

(

)

=cosx

14. (

)

a x a x a cos cos lim 0 − + →

= ? à

(

)

=

a x a x x a x a       − −       + + − → 2 sen . 2 sen 2 lim 0

=

      −       −       + − → 2 . 2 2 sen . 2 2 sen . 2 lim 0 a a a x a

=

      −       −       + − → 2 2 sen . 2 2 sen lim 0 a a a x a

=

x sen −

Logo

(

)

=-senx

15.

a x a x a x − − → sec sec lim

= ? à

a x a x a x − − → sec sec lim

=

a x a x a x − − → cos 1 cos 1 lim

=

a x a x x a a x − − → cos . cos cos cos lim

=

(

x a

)

x a x a a x .cos .cos cos cos lim − − →

=

(

x a

)

x a x a x a a x .cos .cos 2 sen . 2 sen . 2 lim −       −       + − →

=

a x x a x a x a a x cos .cos 1 . 2 . 2 2 sen . 1 2 sen . 2 lim       − −       −       + − →

=

a x x a x a x a a x _cos _._cos 1 . 2 2 sen . 1 2 sen lim       −       −       + →

=

a a a cos . cos 1 . 1 . 1 sen

=

a a a cos 1 . cos sen

=

tga sec

.

a

Logo

a x a x a x − − → sec sec

lim

=

tga sec

.

a

16.

x x x 1 sec lim 2 0 − →

= ? à

x x x 1 sec lim 2 0 − →

=

(

x

)

x x x x cos 1 1 . cos 1 . sen 1 lim 2 2 0 + − →

=

−

2

( )

x x x f cos 1 1 2 − =

=

x x x cos 1 cos 2 −

=

(

x

)

x x cos 1 . 1 cos . 2 − −

=

(

)

(

)

(

x

)

x x x x cos 1 cos 1 . cos 1 . cos 1 1 2 + + − −

=

x 1 1 cos 1 1 2 −

=

sen x 1 1 1 2

(4)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

17.

tgx gx x − − → 1 cot 1 lim 4 π

= ? à

=

tgx tgx x − − → 1 1 1 lim 4 π

=

tgx tgx tgx x − − → 1 1 lim 4 π

=

tgx tgx tgx x − − − → 1 ) 1 .( 1 lim 4 π

=

_x tgx 1 lim 4 − →π

= 1

−

Logo

= -1

18.

x x x 2 3 0 _sen cos 1 lim − →

= ? à

_x x x 2 3 0 _sen cos 1 lim − →

=

(

)

(

)

x x x x x 2 2 0 ₁ _cos cos cos 1 . cos 1 lim − + + − →

=

(

)

(

)

(

x

)(

x

)

x x x x 1 cos .1 cos cos cos 1 . cos 1 lim 2 0 − + + + − →

=

x x x x 1 cos cos cos 1 lim 2 0 + + + →

=

2 3

Logo

x x x 2 3 0 _sen cos 1 lim − →

=

2 3

19.

x x x 1 2.cos 3 sen lim 3 − →π

= ? à

x x x 1 2.cos 3 sen lim 3 − →π

=

(

)

1 cos . 2 1 . sen lim 3 x x x + − →π

=

− 3

( )

x x x f cos . 2 1 3 sen − =

=

(

)

x x x cos . 2 1 2 sen − +

=

x x x x x cos . 2 1 cos . 2 sen 2 cos . sen − +

=

(

)

x x x x x x cos . 2 1 cos . cos . sen . 2 1 cos 2 . sen 2 − + −

=

(

)

[

]

x x x x cos . 2 1 cos 2 1 cos 2 . sen 2 2 − + −

=

[

]

x x x cos . 2 1 1 cos 4 . sen 2 − −

=

(

)(

)

x cox cox x cos . 2 1 . 2 1 . . 2 1 . sen − + − −

=

(

)

1 cos . 2 1 . senx + x −

20.

tgx x x x − − → 1 cos sen lim 4 π

= ? à

tgx x x x − − → 1 cos sen lim 4 π

=

x

(

x

)

cos lim 4 − →π

=

2 2 −

( )

tgx x x x f − − = 1 cos sen

=

x x x x cos sen 1 cos sen − −

=

x x x x cos sen 1 cos sen − −

=

x x x x x cos sen cos cos sen − −

=

₍

₎

x x x x x cos cos sen . 1 cos sen − − −

=

x x x x x sen cos cos . 1 cos sen − − −

=

−cosx

21.

lim

(

3

)

.cossec( )

3 x x

x→ − π

= ? à

xlim→3

(

3−x

)

.cossec(πx)

=

0.∞

( ) (

x 3 x

)

.cossec( x) f = − π

=

(

) ( )

x x π sen 1 . 3−

=

(

x

)

x π π− − sen 3

=

(

x

)

x π π − − 3 sen 3

=

₍

₎

(

x

)

x − − 3 . 3 sen . 1 π π π π

=

(

)

(

x

)

x π π π π π − − 3 3 sen . 1

à

lim

(

3

)

.cossec( ) 3 x x x→ − π

=

(

)

(

x

)

x x π π π π π − − → 3 3 sen . 1 lim 3

=

π

1

22. lim

.

sen(

1 )

x

x→∝

= ? à

) 1 sen( . lim x x x→∝

=

∞.0 x x x 1 1 sen lim       →∝

=

1 sen lim 0 = → t t t

à Fazendo

  → +∞ → = 0 1 t x x t

(5)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

23.

1 sen

.

3 sen

.

2

1 sen

sen

.

2 lim

₂ 2 6

−

+

−

+

→

_x

x

x π

= ? à

1 sen . 3 sen . 2 1 sen sen . 2 lim 2 2 6 − + − + → x x x x x π

=

x x x 1 sen sen 1 lim 6− + + →π

=

6 sen 1 6 sen 1 π π + − +

=

2 1 1 2 1 1 + − +

=

−

3 à

( )

1 sen . 3 sen . 2 1 sen sen . 2 2 2 + − − + = x x x x x f

=

(

)

(

sen 1

)

. 2 1 sen 1 sen . 2 1 sen −       ₋ +       ₋ x x x x

=

(

)

(

sen

1 )

1 sen

−

+

x

=

x x sen 1 sen 1 + − +

24. (

)

      − → 1 . 2 lim 1 x tg x x π

= ? à

(

)

      − → 1 . 2 lim 1 x tg x x π

=

0 .

∞

à

( ) (

)

      − = 2 . 1 x tg x x f π

=

(

)

      − − 2 2 cot . 1 x g π πx

=

(

)

      − − 2 2 1 x tg x π π

=

(

)

      − − 2 2 2 . 1 . 2 x tg x π π π π

=

( )

x x tg −       − 1 . 2 2 2 2 π π ππ

=

      −       − 2 2 2 2 2 x x tg π π π ππ

à

(

)

      − → 1 . 2 lim 1 x tg x x π

=

      −       − → 2 2 2 2 2 lim 1 x x tg x π π π ππ

=

( )

t t tg t 0 lim 2 → π

₌

π

2 Fazendo uma mudança de variável,

temos :

   → → − = 0 1 2 t x x x t π π

25. ( )

x x x senπ 1 lim 2 1 − →

= ? à

( )

x x x senπ 1 lim 2 1 − →

=

(

)

(

x

)

x x x π π π π π − − + → .sen 1 lim 1

=

π

2 ( )

x x x f π sen 1− 2 =

=

(

₍

)(

₎

)

x x x π π − + − sen 1 . 1

=

₍

₎

(

x

)

x x − − + 1 sen 1 π π

=

(

)

(

x

)

x x − − + 1 . sen . 1 π π π π

=

(

)

(

x

)

x x π π π π π − − + sen . 1

26. 









 −

→

g

x

g

x

cot

2 .

cot

₂

lim

0

π

= ? à











 −

→

g

x

g

x

cot

2 .

cot

₂

lim

0

π

=

∞

.

0 ( )

      − = g x g x x f 2 cot . 2 cot π

=

cotg2x.tgx

=

x

tg

tgx

2 =

x tg tgx tgx 2 1 2 −

=

tgx x tg tgx . 2 1 . 2 −

=

2 1−tg2x       − → g x g x x cot 2 .cot 2 lim 0 π

=

2 1 lim 2 0 x tg x − →

=

2 1

(6)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

BriotxRuffini :

1

0

0 ...

0 -1

1

•

1

1 ...

1

1 ...

1

0

28.

x x x x x cos sen 1 2 cos 2 sen lim 4 − − − →π

= ? à

x x x x x cos sen 1 2 cos 2 sen lim 4 − − − →π

=

x

(

x

)

cos . 2 lim 4 − →π

=

2.cos4 π −

=

2 2 . 2 −

=

2 −

( )

x x x x x f sen cos 1 2 cos 2 sen − − − =

=

(

)

x x x x x sen cos 1 1 cos 2 cos sen . 2 2 − − − −

=

x x x x x sen cos 1 1 cos 2 cos . sen . 2 2 − − + −

=

x x x x x sen cos cos 2 cos . sen . 2 2 − −

=

(

)

x x x x x sen cos sen cos . cos . 2 − − −

=

−2.cosx

29. (

)

1

2

1 sen

lim

1

−

→

_x

x

= ? à

(

)

1 1 2 1 sen lim 1 − − − → x x x

=

(

)

(

)

1 1 1 2 . 1 1 sen . 2 1 lim 1 + − − − → x x x x

= 1

( )

1 1 2 1 sen − − − = x x x f

=

(

)

1 1 2 1 1 2 . 1 1 2 1 sen + − + − − − − x x x x

=

(

)

1 1 1 2 . 1 1 2 1 sen − + − − − x x x

=

₍

(

₎

)

1 1 1 2 . 1 . 2 1 sen − + − − x x x

=

(

)

(

)

1 1 1 2 . 1 1 sen . 2 1 − + − − x x x

30.

3 cos . 2 1 lim 3 π π − − → _x x x

= ? à

3 cos . 2 1 lim 3 π π − − → _x x x

=

        ₋         ₋         ₊ → 2 3 2 3 sen . 2 3 sen . 2 lim 3 x x x x π π π π

=

. 2 3 3 sen . 2        π ₊π

=

. 2 3 2 sen . 2         π

=

. 3 sen . 2      π

=

3 2 3 . 2 =

( )

3 cos . 2 1 π − − = x x x f

=

3 cos 2 1 . 2 π −       ₋ x x

=

3 cos 3 cos . 2 π π −       ₋ x x

=

( )

        ₋ −         ₋         ₊ − 2 3 . 2 . 1 2 3 sen . 2 3 sen 2 . 2 x x x π π π

=

        ₋         ₋         ₊ 2 3 2 3 sen . 2 3 sen . 2 x x x π π π

=

        ₋         ₋         ₊ 2 3 2 3 sen . 2 3 sen . 2 x x x π π π

31.

x x x x .sen 2 cos 1 lim 0 − →

= ? à

x x x x .sen 2 cos 1 lim 0 − →

=

x x x sen . 2 lim 3 π →

= 2

(7)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

( )

x x x x f sen . 2 cos 1− =

=

(

)

x

sen

.

sen

2

1

1 −

−

2

=

x x x sen . sen 2 1 1− + 2

=

x

sen

.

sen

.

2

=

x x sen . 2

32.

x x x x ₁ _sen ₁ _sen lim 0 + − − →

= ? à

_x _x x x ₁ _sen ₁ _sen lim 0 + − − →

=

x x x x x 2.sen sen 1 sen 1 lim 0 − + + →

=

₂

_.

₁

1

1 +

=1

( )

x x x x f sen 1 sen 1+ − − =

=

(

₍

₎

)

x x x x x sen 1 sen 1 sen 1 sen 1 . − − + − + +

=

(

)

x x x x x sen 1 sen 1 sen 1 sen 1 . + − + − + +

=

(

)

x x x x sen . 2 sen 1 sen 1 . + + −

=

x x x x sen . 2 sen 1 sen 1+ + −

=

1 . 2 1 1+

= 1

33.

x x x x cos sen 2 cos lim 0 − →

=

1 sen cos lim 0 x x x + →

=

2 2 2 2 ₊

=

2

( )

x x x x f sen cos 2 cos − =

=

₍

(

₎₍

)

₎

x x x x x x x sen cos . sen cos sen cos . 2 cos + − +

=

(

)

x x x x x 2 2 sen cos sen cos . 2 cos − +

=

(

)

x x x x 2 cos sen cos . 2 cos +

=

(

)

x x x x 2 cos sen cos . 2 cos +

=

1 sen cosx+ x

=

2 2 2 2 +

= 2

34.

3 sen . 2 3 lim 3 π π − − → _x x x

= ? à

3 sen . 2 3 lim 3 π π − − → _x x x

=

3 sen 2 3 . 2 lim 3 π π −         − → _x x x

=

3 sen 3 sen . 2 lim 3 π π π −       ₋ → _x x x

=

3 2 3 cos . 2 3 sen . 2 lim 3 π π π π −                         +             − → _x x x x

=

3 3 2 3 3 cos . 2 3 3 sen . 2 lim 3 π π π π −                         +             − → x x x x

=

(

)

3 3 . 1 6 3 cos . 6 3 sen . 2 lim 3 x x x x − −           +       − → π π π π

35. ?