1. 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→
= ? à
x
x
x sen
lim
0→
=
0
0
, é uma indeterminação.
x
x
x sen
lim
0→
=
x
xx sen
1
lim
0→
=
x
x
x
sen
lim
1
0→
= 1 logo
x
x
x sen
lim
0→
= 1
2.
x
x
x
4sen
lim
0→
= ? à
x
x
x
4sen
lim
0→
=
0
0
à
x
x
x 4
4sen
.4lim
0→
= 4.
y
y
y
sen
lim
0→
=4.1= 4 logo
x
x
x
4sen
lim
0→
=4
3.
x
x
x 2
5sen
lim
0→
= ? à =
→ x
x
x 5
5sen
.
2
5
lim
0
=
→ y
y
y
sen
.
2
5
lim
0 2
5
logo
x
x
x 2
5sen
lim
0→
=
2
5
4.
nx
mx
x
sen
lim
0→
= ? à
nx
mx
x
sen
lim
0→
=
mx
mx
n
m
x
sen
.lim
0→
=
n
m
.
y
y
y
sen
lim
0→
=
n
m
.1=
n
m
logo
nx
mx
x
sen
lim
0→
=
n
m
5.
x
x
x 2sen
3sen
lim
0→
= ? à
x
x
x 2sen
3sen
lim
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
2sen
lim
3
3sen
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
x
x
x 2sen
3sen
lim
0→
=
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
=
→ xx
x
x
1
.
cos
sen
lim
0
xx
x
x cos
1
.
sen
lim
0→
=
xx
x
xx cos
1
lim.
sen
lim
00 →→
= 1 Logo =
→ x
tgx
x 0
lim 1
8.
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
= ? à
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
=
0
0
àFazendo



→
→
−=
0
1
,12
t
x
at à
( )
t
ttg
t 0
lim
→
=1
logo
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
=1

2. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
2
9.
xx
xx
x 2sen
3sen
lim
0 +
−
→
= ? à
xx
xx
x 2sen
3sen
lim
0 +
−
→
=
0
0
à ( )
xx
xx
xf
2sen
3sen
+
−
= =






+






−
x
x
x
x
x
x
5sen
1.
3sen
1.
=






+






−
x
x
x
x
x
x
.5
5sen
.51.
.3
3sen
.31.
=
x
x
x
x
.5
5sen
.51
.3
3sen
.31
+
−
à
0
lim
→x
x
x
x
x
.5
5sen
.51
.3
3sen
.31
+
−
=
51
31
+
−
=
6
2−
=
3
1
− logo
xx
xx
x 2sen
3sen
lim
0 +
−
→
=
3
1
−
10. 30
sen
lim
x
xtgx
x
−
→
= ? à 30
sen
lim
x
xtgx
x
−
→
=
xx
x
xx
x
x cos1
1
.
sen
.
cos
1
.
sen
lim 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
sen
lim
x
xtgx
x
−
→
=
2
1
11. 30
sen11
lim
x
xtgx
x
+−+
→
=? à
xtgxx
xtgx
x sen11
1
.
sen
lim 30 +++
−
→
=
xtgxxx
x
xx
x
x sen11
1
.
cos1
1
.
sen
.
cos
1
.
sen
lim 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
.
sen
3
+++
−
30
sen11
lim
x
xtgx
x
+−+
→
=
4
1
12.
ax
ax
ax −
−
→
sensen
lim = ? à
ax
ax
ax −
−
→
sensen
lim =





 −





 +





 −
→
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 −
−
→
sensen
lim = cosa

3. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
3
13.
( )
a
xax
a
sensen
lim
0
−+
→
= ? à
( )
a
xax
a
sensen
lim
0
−+
→
=
1
2
cos.
.
2
.2
2
sen2
lim





 ++





 −





 −+
→
xax
ax
xax
aa
=
1
2
2
cos.
.
2
.2
2
sen2
lim





 +












→
ax
a
a
aa
= xcos Logo
( )
a
xax
a
sensen
lim
0
−+
→
=cosx
14.
( )
a
xax
a
coscos
lim
0
−+
→
= ? à
( )
a
xax
a
coscos
lim
0
−+
→
=
a
xaxxax
a





 −−





 ++
−
→
2
sen.
2
sen2
lim
0
=





 −





 −





 +
−
→
2
.2
2
sen.
2
2
sen.2
lim
0 a
aax
a
=





 −





 −





 +
−
→
2
2
sen
.
2
2
senlim
0 a
a
ax
a
= xsen− Logo
( )
a
xax
a
coscos
lim
0
−+
→
=-senx
15.
ax
ax
ax −
−
→
secsec
lim = ? à
ax
ax
ax −
−
→
secsec
lim =
ax
ax
ax −
−
→
cos
1
cos
1
lim =
ax
ax
xa
ax −
−
→
cos.cos
coscos
lim =
( ) axax
xa
ax cos.cos.
coscos
lim
−
−
→
=
( ) 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 −
−
→
secsec
lim = atga sec.
16.
x
x
x sec1
lim
2
0 −→
= ? à
x
x
x sec1
lim
2
0 −→
=
( )xxx
xx
cos1
1
.
cos
1
.
sen
1
lim
2
20
+
−
→
= 2−
( )
x
x
xf
cos
1
1
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
+
−

4. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
4
17.
tgx
gx
x −
−
→ 1
cot1
lim
4
π
= ? à
tgx
gx
x −
−
→ 1
cot1
lim
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
=
tgxx
1
lim
4
−
→
π
= 1− Logo
tgx
gx
x −
−
→ 1
cot1
lim
4
π
= -1
18.
x
x
x 2
3
0 sen
cos1
lim
−
→
= ? à
x
x
x 2
3
0 sen
cos1
lim
−
→
=
( )( )
x
xxx
x 2
2
0 cos1
coscos1.cos1
lim
−
++−
→
=
( )( )
( )( )xx
xxx
x cos1.cos1
coscos1.cos1
lim
2
0 +−
++−
→
=
x
xx
x cos1
coscos1
lim
2
0 +
++
→
=
2
3
Logo
x
x
x 2
3
0 sen
cos1
lim
−
→
=
2
3
19.
x
x
x cos.21
3sen
lim
3
−→
π
= ? à
x
x
x cos.21
3sen
lim
3
−→
π
=
( )
1
cos.21.sen
lim
3
xx
x
+
−
→
π
= 3−
( )
x
x
xf
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
cossen
lim
4
π
= ? à
tgx
xx
x −
−
→ 1
cossen
lim
4
π
= ( )x
x
coslim
4
−
→π
=
2
2
−
( )
tgx
xx
xf
−
−
=
1
cossen
=
x
x
xx
cos
sen
1
cossen
−
−
=
x
x
xx
cos
sen
1
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−
→
= ? à ( ) )sec(cos.3lim
3
xx
x
π−
→
= ∞.0
( ) ( ) )sec(cos.3 xxxf π−= =( )
( )x
x
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
à ( ) )sec(cos.3lim
3
xx
x
π−
→
=
( )
( )x
xx
ππ
πππ
−
−→
3
3sen.
1
lim
3
=
π
1
22. )
1
sen(.lim
x
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
5
23.
1sen.3sen.2
1sensen.2
lim 2
2
6 +−
−+
→ xx
xx
x π
= ? à
1sen.3sen.2
1sensen.2
lim 2
2
6 +−
−+
→ xx
xx
x π
=
x
x
x sen1
sen1
lim
6
+−
+
→π
=
6
sen1
6
sen1
π
π
+−
+
=
2
1
1
2
1
1
+−
+
= 3− à ( )
1sen.3sen.2
1sensen.2
2
2
+−
−+
=
xx
xx
xf =
( )
( )1sen.
2
1
sen
1sen.
2
1
sen
−





−
+





−
xx
xx
=
( )
( )1sen
1sen
−
+
x
x
=
x
x
sen1
sen1
+−
+
24. ( ) 





−
→ 2
.1lim
1
x
tgx
x
π
= ? à ( ) 





−
→ 2
.1lim
1
x
tgx
x
π
= ∞.0 à ( ) ( ) 





−=
2
.1
x
tgxxf
π
=
( ) 





−−
22
cot.1
x
gx
ππ
=
( )






−
−
22
1
x
tg
x
ππ
=
( )






−
−
22
2
.1.
2
x
tg
x
ππ
π
π
=
( )x
x
tg
−






−
1.
2
22
2
π
ππ
π =






−






−
22
22
2
x
x
tg
ππ
ππ
π à
( ) 





−
→ 2
.1lim
1
x
tgx
x
π
=






−






−
→
22
22
2
lim
1
x
x
tg
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
x
t
ππ
25.
( )x
x
x πsen
1
lim
2
1
−
→
= ? à
( )x
x
x πsen
1
lim
2
1
−
→
=
( )
( )x
x
x
x
ππ
πππ
−
−
+
→ sen.
1
lim
1
=
π
2
( )
x
x
xf
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
= ? à 





−
→
xgxg
x 2
cot.2cotlim
0
π
= 0.∞
( ) 





−= xgxgxf
2
cot.2cot
π
= tgxxg .2cot =
xtg
tgx
2
=
xtg
tgx
tgx
2
1
2
−
=
tgx
xtg
tgx
.2
1
.
2
−
=
2
1 2
xtg−






−
→
xgxg
x 2
cot.2cotlim
0
π
=
2
1
lim
2
0
xtg
x
−
→
=
2
1
27.
x
xx
x 2
3
0 sen
coscos
lim
−
→
= 11102
2
1 ...1
lim
tttt
t
t +++++
−
→
=
12
1
−
( )
x
xx
xf 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 −=

6. 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
12cos2sen
lim
4
−
−−
→π
= ? à
xx
xx
x sencos
12cos2sen
lim
4
−
−−
→π
= ( )x
x
cos.2lim
4
−
→π
=
4
cos.2
π
− =
2
2
.2− =
2−
( )
xx
xx
xf
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
1sen
lim
1
−−
−
→
x
x
x
= ? à
( )
112
1sen
lim
1 −−
−
→ x
x
x
=
( )
( ) 1
112
.
1
1sen
.
2
1
lim
1
+−
−
−
→
x
x
x
x
= 1
( ) ( )
112
1sen
−−
−
=
x
x
xf =
( )
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.21
lim
3
ππ
−
−
→ x
x
x
= ? à
3
cos.21
lim
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
x
xf =
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.
2cos1
lim
0
−
→
= ? à
xx
x
x sen.
2cos1
lim
0
−
→
=
x
x
x
sen.2
lim
3
π
→
= 2

7. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
7
( )
xx
x
xf
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 −−+→
= ? à
xx
x
x sen1sen1
lim
0 −−+→
=
x
x
xx
x sen.2
sen1sen1
lim
0
−++
→
=
1.2
11+
=1
( )
xx
x
xf
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
2cos
lim
0 −→
=
1
sencos
lim
0
xx
x
+
→
=
2
2
2
2
+ = 2
( )
xx
x
xf
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.23
lim
3
ππ
−
−
→ x
x
x
= ? à
3
sen.23
lim
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
3
cos.
6
3
sen.2
lim
3
x
xx
x
−−











 +





 −
→
π
ππ
π
35. ?