LIMITES TRIGONOMÉTRICOS RESOLVIDOS

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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→
= ? 
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→
= ? 
x
x
x
4sen
lim
0→
=
0
0
 
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→
= ? =
→ 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→
= ? 
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→
= ? 
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
→
= ? 
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 ? =
→ x
tgx
x 0
lim
0
0
 =
→ 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
= ? 
( )
1
1
lim
2
2
1
−
−
→ a
atg
a
= 
0
0
 Fazendo 


→
→
−=
0
1
,1
2
t
x
at 
( )
t
ttg
t 0
lim
→
=1 
logo 
( )
1
1
lim
2
2
1
−
−
→ a
atg
a
=1 
Limites Trigonométricos Resolvidos 
Sete páginas e 34 limites resolvidos 
 
 
2 
9. 
xx
xx
x 2sen
3sen
lim
0 +
−
→
= ? 
xx
xx
x 2sen
3sen
lim
0 +
−
→
= 
0
0
 ( )
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
+
−
 
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
−
→
= ? 
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
+−+
→
=? 
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 = ? 
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 
Limites Trigonométricos Resolvidos 
Sete páginas e 34 limites resolvidos 
 
 
3 
13. 
( )
a
xax
a
sensen
lim
0
−+
→
= ? 
( )
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
−+
→
= ? 
( )
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 = ? 
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
−
→
= ? 
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
+
−
 
Limites Trigonométricos Resolvidos 
Sete páginas e 34 limites resolvidos 
 
 
4 
17.tgx
gx
x −
−
→ 1
cot1
lim
4
pi
= ? 
tgx
gx
x −
−
→ 1
cot1
lim
4
pi
= 
tgx
tgx
x −
−
→ 1
1
1
lim
4
pi
 =
tgx
tgx
tgx
x −
−
→ 1
1
lim
4
pi
= 
tgx
tgx
tgx
x −
−−
→ 1
)1.(1
lim
4
pi
=
tgxx
1
lim
4
−
→
pi
= 1− Logo 
tgx
gx
x −
−
→ 1
cot1
lim
4
pi
= -1 
18. 
x
x
x 2
3
0 sen
cos1
lim
−
→
= ? 
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
−→
pi
= ? 
x
x
x cos.21
3sen
lim
3
−→
pi
= 
( )
1
cos.21.sen
lim
3
xx
x
+
−
→
pi
= 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
pi
= ? 
tgx
xx
x −
−
→ 1
cossen
lim
4
pi
= ( )x
x
coslim
4
−
→pi
=
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
pi−
→
= ? ( ) )sec(cos.3lim
3
xx
x
pi−
→
= ∞.0 
( ) ( ) )sec(cos.3 xxxf pi−= = ( ) ( )xx pisen
1
.3 − = ( )x
x
pipi −
−
sen
3
= ( )x
x
pipi −
−
3sen
3
= ( )
( )x
x
−
−
3.
3sen.
1
pi
pipipi
= 
( )
( )x
x
pipi
pipipi
−
−
3
3sen.
1
 ( ) )sec(cos.3lim
3
xx
x
pi−
→
= ( )
( )x
xx
pipi
pipipi
−
−→
3
3sen.
1
lim
3
=
pi
1
 
22. )
1
sen(.lim
x
x
x→∝
= ? )
1
sen(.lim
x
x
x→∝
= 0.∞ 
x
x
x 1
1
sen
lim



→∝
= 1
sen
lim
0
=
→ t
t
t
 Fazendo 


→
+∞→
=
0
1
t
x
x
t 
Limites Trigonométricos Resolvidos 
Sete páginas e 34 limites resolvidos 
 
 
5 
23. 
1sen.3sen.2
1sensen.2
lim
2
2
6 +−
−+
→ xx
xx
x pi
= ? 
1sen.3sen.2
1sensen.2
lim
2
2
6 +−
−+
→ xx
xx
x pi
=
x
x
x sen1
sen1
lim
6
+−
+
→pi
=
6
sen1
6
sen1
pi
pi
+−
+
= 
2
1
1
2
1
1
+−
+
= 3− ( )
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
pi
= ? ( ) 


−
→ 2
.1lim
1
x
tgx
x
pi
= ∞.0 ( ) ( ) 


−=
2
.1
x
tgxxf
pi
= 
( ) 


−−
22
cot.1
x
gx
pipi
 = 
( )



−
−
22
1
x
tg
x
pipi
=
( )



−
−
22
2
.1.
2
x
tg
x
pipi
pi
pi
 = 
( )x
x
tg
−



−
1.
2
22
2
pi
pipi
pi =



−



−
22
22
2
x
x
tg
pipi
pipi
pi 
( ) 


−
→ 2
.1lim
1
x
tgx
x
pi
=



−



−
→
22
22
2
lim
1
x
x
tg
x
pipi
pipi
pi = ( )
t
ttg
t 0
lim
2
→
pi =
pi
2
 Fazendo uma mudança de variável, 
temos : 


→
→
−=
0
1
2 t
x
x
x
t
pipi
 
25. ( )x
x
x pisen
1
lim
2
1
−
→
= ? ( )x
x
x pisen
1
lim
2
1
−
→
= ( )
( )x
x
x
x
pipi
pipipi
−
−
+
→ sen.
1
lim
1
=
pi
2
 
( )
x
x
xf
pisen
1 2−
= = 
( )( )
( )x
xx
pipi −
+−
sen
1.1
= ( )
( )x
x
x
−
−
+
1
sen
1
pipi
= ( )
( )x
x
x
−
−
+
1.
sen.
1
pi
pipipi
= ( )
( )x
x
x
pipi
pipipi
−
−
+
sen.
1
 
26. 


−
→
xgxg
x 2
cot.2cotlim
0
pi
= ? 


−
→
xgxg
x 2
cot.2cotlim
0
pi
= 0.∞ 
( ) 


−= xgxgxf
2
cot.2cot
pi
 = tgxxg .2cot =
xtg
tgx
2
=
xtg
tgx
tgx
21
2
−
= 
tgx
xtg
tgx
.2
1
.
2
−
=
2
1 2xtg−
 



−
→
xgxg
x 2
cot.2cotlim
0
pi
=
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 −= 
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
−
−−
→pi
= ? 
xx
xx
x sencos
12cos2sen
lim
4
−
−−
→pi
= ( )x
x
cos.2lim
4
−
→pi
= 
4
cos.2
pi
− =
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
= ? 
( )
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
pipi
−
−
→ x
x
x
= ? 
3
cos.21
lim
3
pipi
−
−
→ x
x
x
= 








−








−







 +
→
2
3
2
3
sen
.
2
3
sen.2lim
3 x
x
x
x pi
pi
pi
pi
= 
.
2
33sen.2 






 + pipi
= .
2
3
2
sen.2 






 pi
= .
3
sen.2 

 pi
= 3
2
3
.2 = 
( )
3
cos.21
pi
−
−
=
x
x
xf =
3
cos
2
1
.2
pi
−



−
x
x
=
3
cos
3
cos.2
pi
pi−



−
x
x
 = 
( )








−
−








−







 +
−
2
3
.2.1
2
3
sen.
2
3
sen2.2
x
xx
pi
pipi
= 








−








−







 +
2
3
2
3
sen.
2
3
sen.2
x
xx
pi
pipi
= 








−








−







 +
2
3
2
3
sen
.
2
3
sen.2
x
x
x
pi
pi
pi
 
31. 
xx
x
x sen.
2cos1
lim
0
−
→
= ? 
xx
x
x sen.
2cos1
lim
0
−
→
=
x
x
x
sen.2
lim
3
pi
→
= 2 
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
−−+→
= ? 
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
pipi
−
−
→ x
x
x
= ? 
3
sen.23
lim
3
pipi
−
−
→ x
x
x
= 
3
sen
2
3
.2
lim
3
pipi
−




−
→ x
x
x
= 
3
sen
3
sen.2
lim
3
pi
pi
pi
−



−
→ x
x
x
= 
3
2
3
cos.
2
3
sen.2
lim
3
pi
pipi
pi
−
















+








−
→ x
xx
x
= 
3
3
2
3
3
cos.
2
3
3
sen.2
lim
3
pi
pipi
pi
−















 +







 −
→
x
xx
x
= 
( )
3
3.1
6
3
cos.
6
3
sen.2
lim
3
x
xx
x
−−



 

 +

 −
→ pi
pipi
pi
 
 
35. ?