Limites Trigonometricos Resolvidos PDF

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Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos Usar o limite fundamental e alguns artifícios : x

1. lim

0 sen

x

x

lim x

0 sen

x

2. lim x

0

x

=?



1 = lim = sen x x

x 0

sen 4x =? x

sen 4x =4 x sen 5 x = ? lim0 x 2x

x

lim

x

lim

x 0



lim x

0

1 sen x x sen 4 x x

0 sen

x

=

lim0 x

senx 1 x

0 , é uma indeterminação. 0

x = 1 logo lim =1 x 0

=

0 0



sen x

lim 4.

x 0

sen y sen 4x = 4. lim 0 y 4x y

=4.1= 4

logo

lim x

3.

0

4. lim x

0

sen mx nx

5 sen 5 x lim0 . x 2 5x



=? 

0 sen 2 x

3 2

6.

logo

senmx lim x 0 sennx lim x

7.

0

senmx sennx

=?  =

a

a

2

1 1

=? 

logo lim1 tg a2 a

0

sen mx nx

= lim x

0

5 2

m sen mx . n mx

=

logo lim

x 0

m sen y . lim n y 0 y

sen 3 x 3x lim x 0 sen 2 x 2. 2x

=

m m .1= n n

.

logo lim

x 0

sen mx m = nx n

sen y 3 y 0 y . sen t 2 lim t 0 t

sen 3 x x 0 3x sen 2 x lim x 0 2x

lim

lim

3.

sen mx m. mx lim x 0 sen nx n. nx

=

sen 5 x 5 = 2x 2

sen mx m = lim . mx x 0 n sen nx nx

=

m n

3 .1= 2

Logo

m n

lim tgx ?  x 0 x sen x 1 lim . x 0 x cos x

8. lim1 tg a2

x

5 sen y . y

02

sen 3x sen 3x = x lim lim x 0 sen 2 x x 0 sen 2x x sen 3x 3 = lim x 0 sen 2x 2 sen mx sen mx = lim x lim x 0 sen nx x 0 sen nx x

sen 3 x =?  5. lim x

lim

lim

y

a

2

1 1

sen x tgx 0 tgx lim cos x lim lim  lim x 0 x x 0 x x 0 x 0 0 x sen x 1 tgx = 1 Logo lim = lim0 . lim0 x x x 0 x x cos x 2 x 1 tg a = 0  Fazendo t a 2 1, lim a 1 a2 1 0 t

sen x 1 . cos x x

1 1 0



lim t

0

tg t t

=1

=1

1

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 9. lim0 x sen 3 x = ?  x

x

sen 3x 3. x sen 5x x. 1 5. 5. x lim

x sen 3 x sen 2 x

0x

0

=

sen x x

3



lim

= cos x

x

1 tgx

0

sen x

=

3

x

1 sen x x

0

x

sen x 1 1 cos x 1 cos x . 2 . . cos x 1 cos x x x Logo lim tgx 3sen x = 1 x 0 2 x

11. lim



0

f x

sen 3 x 3.x = 1 3 lim sen 5 x x 0 1 5 1 5. 5.x 1 3.



3

tgx sen x

x

1 tgx

1 senx x

lim x

1 tgx

0

12. lim x a

3

3

sen x sen a x

=

logo

a

=1 2

= sen x . 12 . 1 cos x = x

cos x

x

2

sen x 1 1 cos x 1 sen x 1 sen x 1 . . . . . . = x cos x 1 cos x x cos x x2 x 2 1 cos x

=? 

lim0

tgx sen x x3

x

= 1 tgx 13 sen x .

=

1 1 cos x

=

.

1 1 tgx

1 1 1 1

1

1 1 2 2

4

=1. . . . = 1

1 tgx

1 sen x

1 sen x

=

x = tgx sen . 3 x

1 1 tgx

1 sen x

1

=? 

4

lim

x

x a ) . cos 2 2 . lim x a x a 1 2. 2 2 sen(

1 3

6

2

x

1 sen x x

= 2=

x x cos x x3 x sen x sen x . cos x cos x = sen x3. 1 cos x 3 x . cos x x

sen x 1 sen x 1 1 . . . . x cos x x2 1 cos x 1 tgx 1 sen x

f x

x. 1

sen 5x x

=

2 = lim0 sen x . 1 . sen2 x .

2

lim0

x sen 3x x sen 2x

3

x

tgx

= 0

sen 3x x

= 1

sen x f x

sen 3x sen 2x

1 3.

10. lim tgx 3sen x = ? x

x

0x

x

sen 3x 3. x sen 5x 1 5. 5. x

x. 1 3.

x

lim

sen 2 x

x. 1

a

sen x x

sen a a

2 sen

= lim x

a

x a 2 2.

. cos

x a 2

x a 2

=

x a

= cos a

Logo lim x a

sen x sen a x

a

= cosa

2

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 13. lim sen x a sen x = ?  a

2 sen lim

a

a

lim

a

0

a

2x

. cos

2

0

sen x a a

sen x

= lim a

a

lim

a

0

lim a

15.

0

cos x a a

cos x

sec x sec a lim x a x a

lim x

a

x

2. sen lim x

a

x 2

1

a

16. lim0 x

x2 1 sec x

sec x sec a lim x a x a 2. sen

= lim x

=? 

a

x

.

x

x2 1 1 cos x 1

1 cos 2 x x

2

=

a 2

x

=cosx

. sen

x a 2

a

= sen x

x

=

Logo

= lim

x2 cos x 1 cosx

1 1 . . cos x 1 cos x

a x 2 1

a

=

x2 . cos x 1. 1 cos x

sen .

a x 2

1 sen x

.

.

1 = cos x. cos a

1

1

=

a

2

. cos x 1 cos x 1

1 cos x x

=

a x 2

x

2

=

=

Logo xlima sec x sec a = tga .sec a

= tga .sec a

x2 = lim 0 1 sec x x 0

=

=

sen x

cos a cos x lim cos x. cos a x a x a

x

2

lim x

a

. sen

2 2 a . cos x. cos a

1 cos x . cos a

sen a 1 . cos a cos a

=

=

x2 f x

sen x

x

2 sen

1 1 lim cos x cos a x a x a

x 2

.

a

x

a

a

sen 2.

sen a 1 .1. 1 cos a. cosa

=

=-senx

=? 

cos a cos x a . cos x. cos a

.

a

0

cos x = ?  lim cos x a cos x = lim a 0 a 0 a a a a 2x a sen 2. sen . sen 2 2 2 = lim sen 2 x a . a 0 a a 2 2. 2 2

0

x a x 2 1

. cos

x a 2. 2

a

Logo lim sen x a

= cos x

1

14. lim cos x a a

x a x 2

a 2

.

a 2. 2

a

2 sen

2

.

1 1 cos x . cos x 1 cos x

=

1 sen 2 x x

2

.

1 1 . cos x 1 cos x

3

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 17.

1 cot gx lim 1 tgx x

=?

lim x

4

lim x

= lim x

4

18.

1 cot gx 1 tgx

= lim

3 lim 1 cos x x 0 sen 2 x

tgx tgx

1

x

4

1.(1 tgx) tgx 1 tgx

1

1

tgx 1

= lim tgx = 1

x

4

1 tgx

= 1

Logo lim x

4

1 cot gx 1 tgx

= -1

4 3 lim 1 cos x x 0 sen 2 x

=? 

tgx

4

2

=

1 cos x . 1 cos x cos x lim x 0 1 cos 2 x

=

3 2 1 cos x cos2 x 3 1 cos x . 1 cos x cos x lim 1 cos x Logo = = lim x 0 x 0 x 0 sen 2 x 1 cos x . 1 cos x 1 cos x 2 sen 3 x sen 3 x sen x . 1 2. cos x = 3 =?  = lim lim lim 1 2. cos x 1 2. cos x x 1 x x

lim

19.

3

3

=

3 2

3

sen x. 2 cos 2 x 1 2. sen x. cos x. cos x sen 3x = sen x 2x = sen x . cos 2x sen 2x . cos x = f x 1 2. cos x 1 2. cos x 1 2. cos x 1 2. cos x sen x. 2 cos 2 x 1 2 cos 2 x sen x. 4 cos 2 x 1 sen x. 1 2. cox . 1 2. cox = sen x. 1 2. cos x = = 1 2. cos x 1 2. cos x 1 2.cos x 1 sen x cos x 1 tgx 4

20. xlim

sen x cos x 1 tgx

=? 

sen x cos x 2 = lim cos x = 2 x 1 tgx 4 4 sen x cos x sen x cos x sen x cos x = = sen x cos x sen x 1. sen x cos x 1 cos x cos x cos x

lim

x

sen x cos x = sen x 1 cos x sen x cos x cos x = cos x . 1 cos x sen x 21. xlim 3 x . cos sec( x) = ?  lim 3 x . cos sec( x) = 0. x f x

=

3 x . cos sec( x)

1 . sen 3 x 3 x



lim x

1 x

1 x

= lim0 t

1 x. sen x

=3

=

lim 3 x . cos sec( x) x 3

1 22. lim x. sen( ) = ?  x x sen

=

3

3

f x

=

sen t t

3 sen

x x

= lim

x 3

=

3 x sen 3 x

1 . sen 3 x 3 x

=

=

1 . sen 3 x .3 x

=

1

1 lim x. sen( ) = .0 x

x

1



Fazendo t

1

x

x

t

0

4

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 23. lim x

6

2. sen 2 x sen x 1 =?  2. sen2 x 3. sen x 1

1 2 1 1 2

1

= 3 



2

x 1

1 x . cot g

x 2

2

x

lim 1 x 1

1 x

= tg

tg

2 lim 1 x .tg x

1

= lim

x x x t

t

temos :

2

= tg

2

1 x2 sen x

x

tg t lim t 0 t

2

1

sen

x

0

f x cot g2 x. cot g

=

1 x sen x

lim x

= 1 x .1 x =

26. lim cot g 2x. cot g

2

x 2

x 2 2

.1

x

= tg

1 sen

=

6

sen x 1 1 sen x = 1 sen x sen x 1

=

1 x .tg

6

2 2

=



x 2 x 2

2

=

Fazendo uma mudança de variável,

1 0

1 x =?  25. lim x 1 sen x

f x

6

1 sen

=

2

x tg 2 2 x 2 2

x 1

x

2

2 x 2

1

f x



2

x 2

2

3. sen x

1 sen x 1 sen x

= lim

1 sen x . sen x 1 2 1 sen x . sen x 1 2

= 0.

.1 x .

=2

x 2

2

x 2

x . tg

sen x 1

2. sen 2 x

6

2. sen 2 x sen x 1 = 2. sen 2 x 3. sen x 1

f x

24. lim 1 x .tg x = ?

2

2. sen x

lim

x

1 sen 1

1

x

1 x x . sen x

2

=

1 x x . sen .1 x

=

x x

=

lim cot g 2x. cot g

x =? 

2

x

0

1 x . sen x x

x =

2

.0

2 2 = cot g 2 x.tgx = tgx = tgx = tgx. 1 tg x = 1 tg x 2tgx tg2 x 2.tgx 2

x

2

= lim

1 tg 2 x lim cot g 2 x. cot g

x

27. lim cos x x

f x t

2

0

2.3

0

3

cos x

2

sen x cos x 3 cos x sen 2 x

cos x

6

x

cos x

= lim

1 tg 2 x

=1

2

x 0

2 2

= lim t

11

=t

3

t

t t 2

1 t 12

x t

0 1

t ... t 10

2

t 11

=

1 12

2

=

t .1 t 1 t .1 t t 6

t

cos x ,

2

... t 10 t 11 t 12

cos 2 x

=

1 t t

t 2

2

... t

10

t

11

, sen 2 x 1 t 12

5

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos BriotxRuffini : 1 0 0 1 • 1 1 1 1 1

... ... ...

0 1 1

-1 1 0

sen 2 x cos 2 x 1 =?  28. lim x

lim

cos x sen x

4

x

4

sen 2 x cos 2 x 1 cos x sen x

= lim x

= 2. cos

2. cos x

4

2 = = 2.

4

2

2 2 cos 2 x 1 sen x

sen 2 x cos 2 x 1 2. sen x cos x = cos x sen x cos x 2. sen x. cos x 2 cos 2 x 2. cos x . cos x = cos x sen x cos x sen f x

29. lim1 x

sen x 1 =?  2x 1 1

sen x 1

f x

=

2x 1 1

lim

x 1

sen x 1 2x 1 1

sen x x

sen x 1 2x 1 1 2x 1 1

.

2x 1 1

1

=

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

=

= 2. cos x

1 sen x 1 2x 1 1 . =1 = lim . x 12

=

x 1

sen x 1 2x 1 1

1

2x 1 1 1

.

sen x 1 2 x 1 1 . = 2. x 1 1

=

1 sen x 1 2x 1 1 . . x 1 2 1

1 2. cos x =? 30. lim x

2. sen

3

3

x

3 2

2. sen

2

3 .= 2. sen

2

1 2. cos x

f x

lim x

x

3



2.

=

2

1 2

x . sen

3 2

1 cos 2 x =?  31. lim x

0

x . sen x

.

3

3 = 2.

=

. x

3

x x

x

3

. sen

2

=

3 2

x

= 1. 2.

3

3 2

sen

2. 2 sen

cos x

3 x

3

3

2

2 . cos

3 2

2

x

2

=

x

3 2

3

3 . = 2. sen

= 2. sen 3 2

= xlim 2. sen 3

x

cos x

x

3

3

1 2. cos x

x

3

sen

x

3 2

x

. x

3 2

lim x

0

1 cos 2 x 2. sen x = lim = x .sen x x x

2

3

6

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 2 2 1 1 2 sen 2 x 2. sen x 2. sen x 1 cos 2x 1 1 2 sen x = = = = x . sen x x . sen x x x. sen x x .sen x

f x

32. lim0 x

x 1 sen x

1 sen x

x

= ?  lim0 x

1 sen x

1 sen x

= lim x

0

1 sen x 1 sen x 2. sen x x

=

1 1 2.1

=1 x. 1 sen x 1 sen x x. 1 sen x 1 sen x x = = 1 sen x 1 sen x 1 sen x 1 sen x 1 sen x 1 sen x 1 sen x 1 sen x 1 sen x x . 1 sen x = =1 1 = 1 sen x 2 . sen x 2.1 2. x

=

f x

cos x sen x 2 2 cos 2 x = 2 = = lim 1 2 2 cos x sen x x 0 cos 2x . cos x sen x cos 2 x cos 2x . cos x sen x = = f x cos x sen x . cos x sen x cos x sen x cos 2 x sen 2 x cos 2x . cos x sen x cos x sen x 2 2 = 2 = = cos 2 x 1 2 2

33. lim0 x

2.

34. lim x

3

3 2. sen x x

2. sen

=?



lim x

3 x

3

. cos

2

2. sen x 3

x

3

x

3

3x . cos 6 1.

3

= lim x

3

2. sen

2

= lim x

3

lim

2. sen x

x

3

lim x

3

3 2 x

3

3x 3 . cos 2

=

cos 2 x. cos x sen x cos 2x

sen x

2. sen

= lim x

3

3

sen x

3

x

=

= 3

3x 3 2

=

3x 3

3x 6

3x 3

35. ?

7...