à x x x 4 limsen →0 = 0 0 à x x x 4 4 .sen 4 lim →0 = 4

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(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

lim 1

0 =

x x

x

limsen 1

0

= 1 logo

x x

x sen

lim

0 = 1

2. x

x

x

4 limsen

0 = ? à

x x

x

4 limsen

0 =

0 0 à

x x

x 4

4 .sen 4 lim

0 = 4.

y y

y

limsen

0 =4.1= 4 logo

x x

x

4 limsen

0 =4

3. x

x

x 2

5 limsen

0 = ? à =

x

x

x 5

5 .sen 2 lim5

0 =

y

y

y

.sen 2 lim 5

0 2

5 logo

x x

x 2

5 limsen

0 =

2 5

4. nx

mx

x

limsen

0 = ? à

nx mx

x

limsen

0 =

mx mx n

m

x

.sen lim

0 =

n m.

y y

y

limsen

0 =

n m.1=

n

m logo

nx mx

x

limsen

0 =

n m

5. x

x

x sen2 3 limsen

0 = ? à

x x

x sen2 3 limsen

0 = =

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 limsen

3 3 limsen

0

0 =

x x x

x

x

x . .1

2 3 limsen

limsen

0

0 =

t t y

y

t

y =

2

3 logo

x x

x sen2 3 limsen

0 =

2 3

6. sennx senmx

x 0

lim = ? à

nx mx

x sen

limsen

0 =

x nx x

mx

x sen

sen lim

0 =

nx n nx

mx m mx

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 lim sen

0

x x

x

x cos

. 1 lim sen

0 =

x x

x

x

x cos

lim 1 sen . lim

0

0

= 1 Logo =

x tgx

x 0

lim 1

8. ( )

1 lim 1

2 2

1

a a tg

a = ? à ( )

1 lim 1

2 2

1

a a tg

a =

0

0 à Fazendo

= 0

, 1

2 1 t a x

t à ( )

t t tg

t 0

lim =1

logo ( )

1 lim 1

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 lim sen

0 +

= ? à

x x

x x

x sen2

3 lim sen

0 +

=

0

0 à ( )

x x

x x x

f sen2

3 sen +

= =

 +

 − x x x

x x x

5 1 sen .

3 1 sen .

=

 +

 − 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

+

= 1 5 3 1

+

=

6

2

= 3

1 logo

x x

x x

x sen2

3 lim sen

0 +

=

3

1

10. 3

0

lim sen x

x tgx

x

= ? à 3

0

lim sen x

x tgx

x

=

x x x x x

x

x 1 cos

. 1 .sen cos . 1 limsen

2 2

0 +

=

2 1

( ) 3sen

x x x tgx

f

= = 3

cos sen 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

lim sen x

x tgx

x

=

2 1

11. 3

0

sen 1 lim 1

x

x tgx

x

+

+

=? à

x tgx

x x tgx

x 1 1 sen

. 1 lim sen

0 3 + + +

=

x x tgx

x x x x

x

x 1 1 sen

. 1 cos 1 . 1 .sen cos . 1 limsen

2 2

0 + + + +

=

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

1 =

4 1

( ) 1 31

x

senx x tgx

f + +

= =

x tgx

x x tgx

sen 1 1

. 1 sen 1 1

3 + + +

+ =

x x tgx

x tgx

sen 1 1

. 1 sen

3 + + +

0 3

sen 1 lim 1

x

x tgx

x

+

+

=

4 1

12. x a

a x

a

x

sen

limsen = ? à

a x

a x

a

x

sen

limsen =

 −

 +

 −

. 2 2

cos 2 2 . sen 2

lim x a

a x a x

a

x =

1 cos 2 . . . 2

2 2 ) sen(

2 lim

 +

 −

a x

a x

a x

a

x = cosa Logo

a x

a x

a

x

sen

limsen = cosa

(3)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

13. ( )

a x a x

a

sen limsen

0

+

= ? à ( )

a x a x

a

sen limsen

0

+

=

1 cos 2 . . . 2

2 sen 2 2 lim

+ +

 −

+

x a x

a x

x a x

a

a =

1 2 cos 2 . . . 2 2 sen 2 2 lim

+

a x

a a

a

a = cosx Logo ( )

a x a x

a

sen limsen

0

+

=cosx

14. ( )

a x a x

a

cos limcos

0

+

= ? à ( )

a x a x

a

cos limcos

0

+

=

a

x a x x a x

a

+ +

sen 2 2 .

sen 2 lim

0 =

 −

 −

+

. 2 2

sen 2 2 . sen 2 . 2 lim

0 a

a a

x

a =

 −

 −

+

2 sen 2 2 . sen 2 lim

0 a

a a

x

a = senx Logo

( )

a x a x

a

cos limcos

0

+

=-senx

15. x a

a x

a

x

sec

limsec = ? à

a x

a x

a

x

sec

limsec =

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)a x x a

a

x .cos .cos

cos lim cos

= (x a) x a

x a x a

a

x .cos .cos

sen 2 2 . sen . 2

lim

 −

 +

=

a x x a

x a x

a

a

x cos .cos

. 1

. 2 2 sen 2 1 .

sen 2 . 2 lim

 −

 −

 +

=

a x x

a x a x

a

a

x cos .cos

. 1

2 sen 2 1 .

sen 2 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

limsec =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 lim 1

2 0 2

+

= −2

( )

x x x

f

cos 1 1

2

= =

x x x

cos 1 cos

2

=

( xx)

x cos 1 . 1

cos

2.

= ( ) ( )

( xx)

x x x

cos 1

cos .1 cos . 1 cos 1

1

2 +

+

=

x 1 1

cos 1

1

2 =

x 1 1

sen 1

2

(4)

Limites Trigonométricos Resolvidos

Sete páginas e 34 limites resolvidos

17. tgx

gx

x

1 cot lim1

4

π = ? à

tgx gx

x

1 cot lim1

4

π =

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 lim 1

4

π = 1 Logo

tgx gx

x

1 cot lim1

4

π = -1

18.

x x

x 2

3 0 sen

cos lim1

= ? à

x x

x 2

3 0 sen

cos lim1

= ( )( )

x

x x

x

x 2

2

0 1 cos

cos cos 1 . cos lim 1

+ +

=

( )( )

( x x)( x x) x

x 1 cos .1 cos

cos cos 1 . cos lim 1

2

0 +

+ +

=

x x x

x 1 cos

cos cos lim1

2

0 +

+ +

=

2

3 Logo

x x

x 2

3 0 sen

cos lim1

=

2 3

19. x

x

x 1 2.cos 3 lim sen

3

π = ? à

x x

x 1 2.cos 3 lim sen

3

π = ( )

1 cos . 2 1 . lim sen

3

x x

x

+

π = 3

( ) x

x x

f 1 2.cos 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 lim sen

π4 = ? à

tgx x x

x

1

cos lim sen

π4 = ( x)

x

cos lim

4

π =

2

2

( ) tgx

x x x

f

= 1

cos

sen =

x x

x x

cos 1 sen

cos sen

=

x x

x x

cos 1 sen

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 lim(3 ).cossec( )

3 x x

x π

=0.

( ) (x 3 x).cossec( x)

f = π =(3x) ( ).sen1πx = sen3(πxπx)=sen(33πxπx)= ( ) (3x) x

. 3 sen .

1 π

π π

π =

( )

(π ππx)πx

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 . lim 1

3 =

π 1

22. 1)

sen(

.

limx x

x→∝ = ? à 1)

sen(

. limx x

x→∝ =.0

x x

x 1

sen 1 lim

→∝ = sen 1 lim

0 =

t t

t à Fazendo

+∞

=

0 1

t x t x

Figure

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Referencias

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