Limites Trigonométricos Resolvidos - Marcello Chaves

Prévia do material em texto

Cálculo I 
Profº. Marcello Santos Chaves 
 
1 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
LIMITES TRIGONOMÉTRICOS 
 
 
Fundamental: 1
0
=
→ x
senxLim
x
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2)(
12)(
2
22)(
2
22)(
2
22)(
2)(
:
2)()1
0
0
000
00
00
00
00
=
×=
⋅=
⋅=
⋅=
=
=
→
→
→→→
→→
→→
→→
→→
xfLim
xfLim
x
xsenLimLimxfLim
x
xsenLimxfLim
x
xsenLimxfLim
x
xsenLimxfLim
Solução
x
xsenLimxfLim
x
x
xxx
xx
xx
xx
xx
4
3)(
14
13)(
4
44
3
33
)(
4
44
3
33
)(
4
44
3
33
)(
4
3
)(
4
3)(
:
4
3)()2
0
0
00
00
0
00
00
00
00
00
=
×
×
=
⋅
⋅
=
⋅
⋅
=
⋅
⋅
=
=
=
=
→
→
→→
→→
→
→→
→→
→→
→→
→→
xfLim
xfLim
x
xSenLimLim
x
xSenLimLim
xfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
xSen
xSenLimxfLim
Solução
xSen
xSenLimxfLim
x
x
xx
xx
x
xx
xx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
2 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1)(
11)(
0
11)(
1)(
1)(
)(
)(
:
)()3
0
0
0
00
00
00
00
00
=
×=
⋅=
⋅=
⋅=
=
=
=
→
→
→
→→
→→
→→
→→
→→
xfLim
xfLim
Cos
xfLim
Cosxx
SenxLimxfLim
xCosx
SenxLimxfLim
x
Cosx
Senx
LimxfLim
x
TgxLimxfLim
Solução
x
TgxLimxfLim
x
x
x
xx
xx
xx
xx
xx
( )
( )
( )
( ) ( )
( )
0)(
01)(
11
01)(
10
011)(
1
1)(
1
1)(
1
)(
1
1)(
1
1)(
1
11)(
1)(
:
1)()4
2
2
22
=
×−=
+
⋅−=
+
⋅×−=
+
⋅⋅−=
+⋅
⋅
⋅−=
+⋅
−
=
+⋅
−
=
+⋅
−
=
+
+
⋅
−
=
−
=
−
=
→
→
→
→
→→
→→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
xfLim
Cos
Sen
xfLim
Cos
SenSenLimxfLim
Cos
SenSenLimxfLim
Cos
SenLimxfLim
Cos
CosLimxfLim
Cos
CosLimxfLim
Cos
CosCosLimxfLim
CosLimxfLim
Solução
CosLimxfLim
x
x
x
x
xx
xx
xx
xx
xx
xx
xx
xx
θ
θ
θ
θ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θθ
θ
θ
θ
θ
θθ
θθ
θθ
θ
θθ
θ
θθ
θ
θ
θ
θ
θ
θ
θ
θ
θ
Cálculo I 
Profº. Marcello Santos Chaves 
 
3 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
9)(
19)(
9
99)(
9
99)(
9
99)(
9)(
:
9)()5
0
0
000
00
00
00
00
=
⋅=
⋅=
⋅=
⋅=
=
=
→
→
→→→
→→
→→
→→
→→
xfLim
xfLim
x
xSenLimLimxfLim
x
xSenLimxfLim
x
xSenLimxfLim
x
xSenLimxfLim
Solução
x
xSenLimxfLim
x
x
xxx
xx
xx
xx
xx
3
4)(
14
3
1)(
4
44
3
1)(
4
44
3
1)(
4
44
3
1)(
3
4)(
:
3
4)()6
0
0
0000
000
000
00
00
=
⋅⋅=
⋅⋅=
⋅⋅=
⋅⋅=
=
=
→
→
→→→→
→→→
→→→
→→
→→
xfLim
xfLim
x
xSenLimLimLimxfLim
x
xSenLimLimxfLim
x
xSenLimLimxfLim
x
xSenLimxfLim
Solução
x
xSenLimxfLim
x
x
xxxx
xxx
xxx
xx
xx
7
10)(
17
110)(
7
77
10
1010
)(
7
77
10
1010
)(
7
10
)(
7
10)(
:
7
10)()7
0
0
00
00
0
00
00
00
00
=
⋅
⋅
=
⋅
⋅
=
⋅
⋅
=
=
=
=
→
→
→→
→→
→
→→
→→
→→
→→
xfLim
xfLim
x
xSenLimLim
x
xSenLimLim
xfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
xSen
xSenLimxfLim
Solução
xSen
xSenLimxfLim
x
x
xx
xx
x
xx
xx
xx
xx
b
a
xfLim
b
a
xfLim
bx
bxSenLimbLim
ax
axSenLimaLim
xfLim
b
b
x
bxSen
a
a
x
axSen
LimxfLim
x
bxSen
x
axSen
LimxfLim
bxSen
axSenLimxfLim
Solução
bxSen
axSenLimxfLim
x
x
xx
xx
x
xx
xx
xx
xx
=
⋅
⋅
=
⋅
⋅
=
⋅
⋅
=
=
=
=
→
→
→→
→→
→
→→
→→
→→
→→
)(
1
1)(
)(
)(
)(
)(
:
)()8
0
0
00
00
0
00
00
00
00
Cálculo I 
Profº. Marcello Santos Chaves 
 
4 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
axfLim
axfLim
Cos
axfLim
axCos
Lim
ax
axSenLimaLimxfLim
axCosax
axSen
aLimxfLim
a
a
xaxCos
axSenLimxfLim
xaxCos
axSenLimxfLim
x
axCos
axSen
LimxfLim
x
axTgLimxfLim
Solução
x
axTgLimxfLim
x
x
x
xxxx
xx
xx
xx
xx
xx
xx
=
⋅⋅=
⋅⋅=
⋅⋅=
⋅⋅=
⋅⋅=
⋅=
=
=
=
→
→
→
→→→→
→→
→→
→→
→→
→→
→→
)(
1
11)(
0
11)(
1)(
1)(
1)(
1)(
)(
)(
:
)()9
0
0
0
0000
00
00
00
00
00
00 ( )
( )
( )
( )
0)(
4
1
1
0)(
4
1)(
4
1)(
4
1)(
4
)(
4
)(
114
)(
1:
14
4
1
1
4
1
)(
1
4
1
)(
1
4
1
)(
:
1
4
1
)()10
11
3 3
3 3
11
3
3
11
3
3
11
=
⋅
−
=
⋅=
⋅=
⋅=
=
=
+−
=
→∴−→
−=∴
+
=→
+





 +
=
+





 +
=
+





 +
=
+





 +
=
→
→
→
→→→
→→
→→
→→
→→
−→−→
−→−→
−→−→
−→−→
xfLim
xfLim
Cos
Sen
xfLim
u
Lim
uCos
uSenLimxfLim
uuCos
uSenLimxfLim
u
uCos
uSen
LimxfLim
u
uTgLimxfLim
u
uTgLimxfLim
uxSe
ux
x
uFaça
x
xTg
LimxfLim
x
xTg
LimxfLim
x
xTg
LimxfLim
Solução
x
xTg
LimxfLim
u
u
u
uuu
uu
uu
uu
uu
xx
xx
xx
xx
pi
pi
pi
pipipi
pipi
pipi
pipi
pipi
pi
pipi
pi
pi
Cálculo I 
Profº. Marcello Santos Chaves 
 
5 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
( )
( )
( )
( ) ( )
( )
( ) ( )
( )
( )
( )
( )
0)(
011)(
11
011)(
10
011)(
1
11)(
1
1)(
1
1)(
1
11)(
1
11)(
1
111)(
11)(
11)(
1)(
:
1)()11
0
0
0
0
0000
000
2
000
2
000
22
000
000
000
00
00
00
=
×−×−=
+
⋅−×−=
+
⋅−×−=
+
⋅
⋅−
⋅−=
+⋅
⋅−
⋅−=
+⋅
−
⋅−=
+⋅
−
⋅−=
+⋅
−
⋅−=
+
+
⋅
−
⋅−=
−
⋅−=
−⋅−
=
−
=
−
=
→
→
→
→
→→→→
→→→
→→→
→→→
→→→
→→→
→→→
→→
→→
→→
xfLim
xfLim
xfLim
Cos
Sen
xfLim
xCos
xSenLim
x
xSenLimLimxfLim
xCosx
xSenxSenLimLimxfLim
xCosx
xSenLimLimxfLim
xCosx
xCosLimLimxfLimxCosx
xCosLimLimxfLim
xCos
xCos
x
xCosLimLimxfLim
x
xCosLimLimxfLim
x
xCosLimxfLim
x
xCosLimxfLim
Solução
x
xCosLimxfLim
x
x
x
x
xxxx
xxx
xxx
xxx
xxx
xxx
xxx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
6 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
( )
( )
( )
( ) ( )
( )
( ) ( )
( )
2
1)(
2
11)(
11
11)(
10
1111)(
1
11)(
1
1)(
1
1)(
1
11)(
1
11)(
1
111)(
11)(
11)(
1)(
:
1)()12
0
0
0
0
00000
000
2
2
000
2
2
000
2
22
000
2000
2000
200
200
200
=






−⋅−=
+
−
⋅−=
+
−
⋅××−=
+
−
⋅⋅⋅−=
+⋅⋅
⋅−
⋅−=
+⋅
−
⋅−=
+⋅
−
⋅−=
+⋅
−
⋅−=
+
+
⋅
−
⋅−=
−
⋅−=
−⋅−
=
−
=
−
=
→
→
→
→
→→→→→
→→→
→→→
→→→
→→→
→→→
→→→
→→
→→
→→
xfLim
xfLim
xfLim
Cos
xfLim
xCos
Lim
x
xSenLim
x
xSenLimLimxfLim
xCosxx
xSenxSenLimLimxfLim
xCosx
xSenLimLimxfLim
xCosx
xCosLimLimxfLim
xCosx
xCosLimLimxfLim
xCos
xCos
x
xCosLimLimxfLim
x
xCosLimLimxfLim
x
xCosLimxfLim
x
xCosLimxfLim
Solução
x
xCosLimxfLim
x
x
x
x
xxxxx
xxx
xxx
xxx
xxx
xxx
xxx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
7 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( ) ( )
( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
pi
pi
pipi
pipi
pi
pipi
pipi
pi
pipi
pipipi
pi
pipipi
pipi
pipi
pipi
pi
pi
pi
1)(
1
1)(
3
3
1
)(
3
3
1)(
3
3
1)(
3
3
1)(
3
3
3
3
)(
3
3)(
3)(
13)(
sec3)(
:
sec3)()13
3
3
33
3
3
33
33
33
33
33
33
33
33
33
=
⋅
=
−
−
⋅
=
−
−
⋅
=
−
−⋅
=
−⋅
−⋅
=
−
−
−
−
=
−
−
=
−
−
=
⋅−=
⋅−=
⋅−=
→
→
→→
→
→
→→
→→
→→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
x
xSenLimLim
Lim
xfLim
x
xSenLimxfLim
x
xSenLimxfLim
x
xSenLimxfLim
x
xSen
x
x
LimxfLim
xSen
xLimxfLim
xSen
xLimxfLim
xSen
xLimxfLim
xCoxLimxfLim
Solução
xCoxLimxfLim
x
x
xx
x
x
xx
xx
xx
xx
xx
xx
xx
xx
xx
( )
( )
( )
( )
7
2)(
14
4)(
1122
126)(
4
4122
2
226
)(
4
4122
2
226
)(
4
4122
2
226
)(
4
4432
2
226
)(
4
4432
2
226
)(
432
26)(
:
432
26)()14
0
0
0
00
00
0
00
00
00
00
00
00
=
=
×+
×−
=






⋅+






⋅−
=






⋅+⋅






⋅−⋅
=






⋅+






⋅−
=






⋅⋅+






⋅−
=
⋅+
⋅−
=
+
−
=
+
−
=
→
→
→
→→
→→
→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
xfLim
x
xSenLimLim
x
xSenLimLim
xfLim
x
xSen
x
x
xSen
x
LimxfLim
x
xSen
xx
x
xSen
xx
LimxfLim
x
xSen
xx
x
xSen
xx
LimxfLim
x
x
xSenx
x
x
xSenx
LimxfLim
xSenx
xSenxLimxfLim
Solução
xSenx
xSenxLimxfLim
x
x
x
xx
xx
x
xx
xx
xx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
8 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2
5)(
11
2
5)(
2
2
2
5
2
5
2
5)(
2
2
2
5
2
5
2
5
)(
2
2
2
5
2
5
2
5
)(
2
2
22
52
)(
2
2
22
52
)(
22
52
)(
22
52
)(
22
52
)(
22
52
)(
2
32
2
322
)(
32)(
:
32)()15
0
0
0000
0000
0000
00
200
200
200
200
200
200
=⇒
××=⇒












⋅












⋅=⇒
⇒












⋅
⋅











⋅
=⇒












⋅






⋅
⋅





=⇒
⇒






⋅






⋅






=⇒






⋅






⋅






=






⋅






=






⋅





=












−⋅





−
=






−⋅





−
=





 −
⋅




 +
−
=
−
=
−
=
→
→
→→→→
→→→→
→→→→
→→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
x
xSen
Lim
x
xSen
LimLimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSenxSen
LimxfLim
x
xSenxSen
LimxfLim
x
xSenxSen
LimxfLim
x
xxSenxxSen
LimxfLim
x
xCosxCosLimxfLim
Solução
x
xCosxCosLimxfLim
x
x
xxxx
xxxx
xxxx
xx
xx
xx
xx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
9 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
[ ]
1)(
21)(
2
2
2)(
2
2
4
2
4
)(2
2
2
2
2
2
4
)(
2
2
2
2
2
2
4
)(
22
4
)(
22
4
)(
2
2
4
)(
2
22
)(
12)(
222)(
2121)(
221)(
:
221)()16
0
0
2
2
002
2
00
2
2
2
2
002
2
2
00
2
2
2
00
2
2
2
00
2
2
2
2
00
2
22
00
2
22
00
2
2
00
2
2
00
2
2
00
200
200
−=⇒
−=⇒
⋅−












=⇒
⇒−






⋅






=⇒−






⋅⋅





⋅






=
−






⋅⋅





⋅






=
−
⋅






=
−






=
−





=






−





⋅
=
−−⋅
=
−−
=
−+−
=
+−
=
+−
=
→
→
→→→→
→→→→
→→
→→
→→
→→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
x
xSenLimLim
x
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSen
xx
xSen
LimxfLim
x
xSen
xx
xSen
LimxfLim
x
xSen
xx
xSen
LimxfLim
x
xSen
x
xSen
LimxfLim
x
xSenxSen
LimxfLim
x
xSenxSen
LimxfLim
x
xSenxCosLimxfLim
x
xSenxCosLimxfLim
x
xSenxCosLimxfLim
x
xCosxCosLimxfLim
Solução
x
xCosxCosLimxfLim
x
x
xxxx
xxxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx