Limites trigonometricos

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Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
senx
=1
x→0 x

Usar o limite fundamental e alguns artifícios :

lim

0
x
x
lim
= , é uma indeterminação.
=? à
x →0 sen x
x → 0 sen x
0
x
1
1
x
lim
= lim
=
= 1 logo lim
=1
sen x
x →0 sen x
x →0 sen x
x → 0 sen x
lim
x→0
x
x
sen 4 x
sen 4 x
sen 4 x 0
sen y
à lim 4.
= 4. lim
= ? à lim
==4.1= 4
2. lim
x →0
y →0
x→0
x→0
0
4x
y
x
x
sen 4 x
lim
=4
x→0
x
sen 5 x
5 sen 5 x
5 sen y 5
sen 5 x
= ? à lim .
logo lim
3. lim
= lim .
=
x →0 2 x
x →0 2
y →0 2
x →0 2 x
y
5x
2
1. lim

sen mx
=
x →0
nx

4. lim

sen 3 x
x →0 sen 2 x

5. lim

x→0

logo
senmx
=
sennx



sen y
m
. lim
n y →0 y

=

=

m
m
.1=
n
n

=

5
2sen mx m
=
x →0
nx
n
sen y
sen 3 x
sen 3 x
sen 3 x
lim
3.
lim
sen 3 x
3 y →0 y
3
x →0 3 x
3x = .
lim
= lim x = lim
=.
= .1 =
sen t
sen 2 x 2
x →0 sen 2 x
x → 0 sen 2 x
x→0
sen 2 x
2
lim
lim
2.
x→0 2 x
t →0 t
x
2x
sen 3 x 3
lim
=
x →0 sen 2 x
2
sen mx
sen mx
sen mx
m.
sen mx
x
mx = lim m . mx = m
lim
= lim
= lim
Logo
sen nx
x →0 sen nx
x →0 n sennx
x → 0 sen nx
x →0
n
n.
nx
nx
x

? à lim

=? à

3
2

6. lim

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

logo

logo lim

senmx m
=
x → 0 sennx
n
lim

7.

8.

sen x
0
tgx
tgx
tgx
sen x 1
lim
=?
à lim
=
= lim cos x = lim
.=
à lim
x→ 0 x
x→ 0 x
x→ 0 x
x→ 0
x → 0 cos x x
0
x
tgx
sen x
1
sen x
1
=1
lim
.
= lim
. lim
= 1 Logo lim
x→0
x→ 0
x → 0 cos x
x→ 0 x
x
cos x
x
x → 1
0
tg (t )
tg a 2 − 1
tg a 2 − 1
= ? à lim 2
=
lim
à Fazendo t = a 2 − 1, 
à lim
=1
2
a →1 a − 1
a →1 a − 1
t →0 t
t →0
0


(

)

logo lim

a →1

(

(

)

) =1

tg a 2 − 1
a2 −1

1

Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
9. lim

x →0

x − sen 3 x
x + sen 2 x

= ? àlim

x →0

x − sen 3 x
x + sen 2 x

0
0

=

à f (x ) =

x − sen 3 x
x + sen 2 x

=

 sen 3 x 
x.1 −

x

=
 sen 5 x 
x.1 +

x


sen 3 x 

sen 3 x
sen 3 x
x.1 − 3.

1 − 3.
1 − 3.
3. x 

3. x
3.x = 1 − 3 = −2 = − 1 logo
=
à lim
sen 5 x
sen 5 x
x →0
sen 5 x 
1+ 5
6
3

1 + 5.
1 + 5.
x.1 + 5.

5. x
5. x
5. x 

x − sen 3x
1
lim
=−
x →0 x + sen 2 x
3
1
sen x 1 sen 2 x
1
tgx − sen x
tgx − sen x
10. lim
= ? à lim
= lim
=
.
.
.
3
3
2
x →0
x →0
x→0
x cos x x
1 + cos x 2
x
x
sen x − sen x. cos x
sen x
− sen x
tgx − sen x cos x
sen x.(1 − cos x ) sen x 1 1 − cos x
cos x
..
f (x ) =
=
=
=
=
3
x x 2 cos x
x 3 . cos x
x3
x3
x
sen x 1 1 − cos x 1 + cos x
.
.
.
x x 2 cos x 1+ cos x
tgx − sen x 1
Logo lim
=
x →0
2
x3

11. lim

=

1 + tgx − 1 + sen x

x →0

x

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

=? à

3

lim

tgx − sen x

x →0

x

3

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

lim

f (x ) =
lim

x →0

12.

1 + tgx − 1 + senx
x3

1 + tgx − 1 + sen x
x

3sen x − sen a
lim
x→a
x−a

=

=

1 + tgx − 1 − sen x
x3

.

.

=

=

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

1
1 + tgx + 1 + sen x

111 1
112 2

= 1. . . . =
1

1 + tgx + 1 + sen x

=

=

1
4
tgx − sen x
x3

.

1
1 + tgx + 1 + sen x

1
4

=? à

sen x − sen a
lim
x→a
x−a

x − a . cos x + a 


)
2
2.
lim
x→a
1x−a
2.

2

=

x−a
x+a
2 sen
. cos

2

 2 =
lim
x→a
x−a
2.

2

2 sen(

= 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

a →0

sen ( x + a ) − sen x
a

= ? à lim

a →0

sen ( x + a ) − sen x
a

a
 2x + a 
2 sen  . cos

2
...
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