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Limites Trigonométricos Resolvidos - Marcello Chaves

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xCosx
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

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