Exercícios Resolvidos de Limites Básicos

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 E CONTINUIDADES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( ) ( )( )
( )
( )
( )
( )
( )
x
x
xfLim
x
x
xfLim
x
x
x
xfLim
x
xfLim
xx
xfLim
xx
LimxfLim
xex
LimxfLim
xexe
eLimxfLim
xexe
xexLimxfLim
xexe
xexLimxfLim
xex
xex
e
xexLimxfLim
Solução
e
xex
xfLim
e
eee
e
ee
ee
ee
ee
ee
ee
e
2
)(
4
2)(
2
2
2
1)(
2
1)(
1)(
0
1
 )(
1
 )(
 )(
 )(
 )(
 )(
:
)()2
0
000
0
00
00
00
00
22
00
00
0
=
⇒=⇒⋅=⇒=
×
=
×+
=
×+
=
×+×
=
×+×
−+
=
×+×
−+
=






++
++
×





−+
=
−+
=
→
→→→
→
→→
→→
→→
→→
→→
→→
→
( )
( )
( )
1)(
111)(
1)(
1
1)1()(
1
1)(
1
1)(
:
1
1)()1
1
2
1
2
11
2
11
3
11
3 3
3
11
3
31
=
++=
++=
−
++⋅−
=
−
−
=
−
−
=
=→
−
−
=
→
→
→→
→→
→→
→→
→
ufLim
ufLim
uuLimufLim
u
uuuLimufLim
u
uLimufLim
u
uLimufLim
xuFaça
Solução
x
x
xfLim
u
u
uu
uu
uu
uu
x
Cálculo I 
Profº. Marcello Santos Chaves 
 
2 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( )
( ) ( )
( ) ( )
6)(
1
23)(
1
111111)(
1)(
1
11)(
1)(
1)(
1)(
1)(
1)(
:
1)()3
1
1
2
2
1
2
2
11
2
2
11
32
3
11
22
3
11
3
11
3
11
3
3
11
3
311
−=
×
−=
−
+⋅++
=
−
+⋅++
=
−⋅−
+⋅++⋅−
=
−
+⋅−
=
−
+⋅−
=
+
+
⋅
−
−
=
−
−
=
−
−
=
=→
−
−
=
→
→
→
→→
→→
→→
→→
→→
→→
→→
→→
ufLim
ufLim
ufLim
u
uuuuuLimufLim
uu
uuuuuuLimufLim
uu
uuuuLimufLim
uuu
uuuuLimufLim
uuu
uuu
uuu
uLimufLim
uuu
uLimufLim
uu
uLimufLim
uxFaça
Solução
xx
xLimxfLim
u
u
u
uu
uu
uu
uu
uu
uu
uu
xx
( )
( ) ( )
( )
1)(
111
1)(
1
1)(
11
1)(
1
1)(
11
:
11)()4
0
20
200
200
300
33
3
00
=
++
=
++
=
++⋅−
−
=
−
−
=
−=⇔+=→
−+
=
→
→
→→
→→
→→
→→
ufLim
ufLim
uu
LimufLim
uuu
uLimufLim
u
uLimufLim
uxxuFaça
Solução
x
xLimxfLim
u
u
uu
uu
uu
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
3 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( ) ( )( )
( )
( )
( )
( )
6
3)(
34
32)(
32
32
32
1)(
33
1)(
330
1)(
33
1)(
33
)(
33
33)(
33
33)(
33
3333)(
:
33)()5
0
00
0
0
00
00
00
22
00
00
00
=
×
=⇒⋅=
+
=
++
=
++
=
++⋅
=
++⋅
−+
=
++⋅
−+
=
++
++
⋅
−+
=
−+
=
→
→→
→
→
→→
→→
→→
→→
→→
→→
xfLim
xfLimxfLim
xfLim
xfLim
x
LimxfLim
xx
xLimxfLim
xx
xLimxfLim
xx
xLimxfLim
x
x
x
xLimxfLim
Solução
x
xLimxfLim
x
xx
x
x
xx
xx
xx
xx
xx
xx
( )[ ]
( )
−∞=
∞−
=
⋅+
∞+⋅−⋅
=






⋅+






−⋅
=






+⋅






−⋅
=
+
−
=
+
−
=
+∞→
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
)(
8
)(
028
305)(
128
315
)(
28
35
)(
28
35)(
:
28
35)()6
2
2
2
3
3
xfLim
xfLim
xfLim
x
x
xLimxfLim
x
x
x
x
x
LimxfLim
x
xLimxfLim
Solução
x
xLimxfLim
x
x
x
xx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
4 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
( )
7
5)(
037
025)(
137
125
)(
37
25
)(
37
25)(
:
37
25)()7
3
3
3
3
3
3
3
3
3
3
−
=
⋅+
⋅−−
=






⋅+






⋅−−
=






+⋅






−⋅−
=
+
+−
=
+
+−
=
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
xfLim
xfLim
x
xLimxfLim
x
x
x
x
LimxfLim
x
xLimxfLim
Solução
x
xLimxfLim
x
x
xx
xx
xx
xx
1)(
1
1)(
01
01)(
11
11
)(
11
11
)(
11
11
)(
11
11
)(
1
1)(
:
1
1)()8
2
2
2
2
2
2
2
2
=
=
+
+
=






+






+
=






+⋅














+⋅
=






+⋅






+⋅
=






+⋅






+⋅
=
+
+
=
+
+
=
+∞→
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
xfLim
xfLim
xfLim
x
x
LimxfLim
x
x
x
x
LimxfLim
x
x
x
x
LimxfLim
x
x
x
x
LimxfLim
x
xLimxfLim
Solução
x
xLimxfLim
x
x
x
xx
xx
xx
xx
xx
xx
Cálculo I 
Profº. Marcello Santos Chaves 
 
5 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( )
( )
( ) ( )
( )
0)(
2
0)(
11
0)(
0101
02)(
1111
12
)(
1111
2
)(
1111
2)(
1111
11)(
1111
11)(
1111
11)(
11
1111)(
:
11)()9
22
22
22
22
22
2
2
2
2
22
2
2
2
2
2
2
2
2
22
22
22
22
=
=
+
=
−++
⋅
=
−++
⋅
=








−++⋅
=








−++⋅
=








−⋅+







+⋅
+−+
=
−⋅++⋅
−−+
=






−⋅+





+⋅
−−+
=
−++
−++
⋅−−+=
−−+=
+∞→
+∞→
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
xfLim
xfLim
xfLim
xfLim
xx
xLimxfLim
x
xx
x
xLimxfLim
xx
x
LimxfLim
x
x
x
x
xxLimxfLim
x
x
x
x
xxLimxfLim
x
x
x
x
xxLimxfLim
xx
xx
xxLimxfLim
Solução
xxLimxfLim
x
x
x
x
xx
xx
xx
xx
xx
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)(
11
1)(
101
1)(
111
1)(
111
1)(
1
11
1)(
1
11
1)(
11
1)(
11
)(
11
)(
111)(
1
1)(
1
11)(
1)(
1)(
:
1)()10
2
22
2
2
2
2
2
2
2
2
2
424
2
2
422
22
22
2
2
22
22
22
22
2
2
−=
⇒
+
−
=⇒
+−
−
=⇒
+−
−
=⇒
⇒
+







−⋅
−
=⇒
+
−⋅
−
=⇒
⇒
+






−⋅
−
=⇒
+
−
−
=








+
−
⋅
−
=








+
−
⋅
−−
=








+
−
⋅
−⋅−
=
+⋅−
−⋅−
=
+⋅−
+⋅−
⋅−⋅−=
−⋅−=
−−=
−−=
+∞→
+∞→+∞→+∞→+∞→
+∞→+∞→+∞→+∞→
+∞→+∞→+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
xfLim
xfLimxfLim
x
LimxfLim
x
x
x
LimxfLim
x
x
x
LimxfLim
x
x
x
LimxfLim
x
x
LimxfLim
x
x
x
xLimxfLim
x
x
x
xxxLimxfLim
x
x
x
xxxLimxfLim
xxx
xxxLimxfLim
xxx
xxx
xxxLimxfLim
xxxLimxfLim
xxxLimxfLim
Solução
xxxLimxfLim
x
xxxx
xxxx
xxxx
xx
xx
xx
xx
xx
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 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
+∞=
∞+
=
−
−∞+
=
−
−
=






−⋅






−⋅
=
−
−
=
−
−⋅
=
−
−
=
=→
−
−
=
+∞→
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
)(
3
)(
03
0)(
13
1
)(
13
1
)(
13
1)(
13
1)(
13
1)(
:
13
1)()11
2
2
2
2
2
2
2
3
2
2
2
22
2
xfLim
xfLim
xfLim
x
x
x
LimxfLim
x
x
x
xx
LimxfLim
x
xLimxfLim
x
xxLimxfLim
x
xxLimxfLim
xvFaça
Solução
v
vvLimvfLim
x
x
x
xx
xx
xx
xx
xx
vv
+∞=
∞+
=
−
−∞+
=
−
−
=






−⋅






−⋅
=
−
−
=
−
−⋅
=
−
−
=
=→
−
−
=
+∞→
+∞→
+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
+∞→+∞→
)(
3
)(
03
0)(
13
1
)(
13
1
)(
13
1)(
13
1)(
13
1)(
:
13
1)()11
2
2
2
2
2
2
2
3
2
2
2
22
2
xfLim
xfLim
xfLim
x
x
x
LimxfLim
x
x
x
xx
LimxfLim
x
xLimxfLim
x
xxLimxfLim
x
xxLimxfLim
xvFaça
Solução
v
vvLimvfLim
x
x
x
xx
xx
xx
xx
xx
vv
Cálculo I 
Profº. Marcello Santos Chaves 
 
8 
Marcello Santos Chaves 
Instituto Federal de Educação, Ciência e Tecnologia (IFPA) Belém-PA, Abril de 2011 
 
 
( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
3
8)(
12
32)(
444
148)(
4222
12242)(
42
124)(
422
1224)(
222
124)(
2
144)(
8
14)(
1
1
8
16)(
8
16)(
:
8
16)()12
2
2
2
2
2
2
2
2
22
2
2
22
22
222
22
33
22
22
3
222
22
3
4
22
3
4
22
3
4
22
−=
−
=
++
−××
=
+⋅+
−⋅+⋅+
=
++
−⋅+⋅+
=
++⋅−
−⋅−⋅+⋅+
=
++⋅−
−⋅−⋅+
=
−
−⋅−⋅+
=
−
−⋅−
=
−
−
⋅
−
−
=
−
−
=
−
−
=
→
→
→
→
→→
→→
→→
→→
→→
→→
→→
→→
xfLim
xfLim
xfLim
xfLim
xx
xxLimxfLim
xxx
xxxLimxfLim
xxx
xxLimxfLim
x
xxLimxfLim
x
xLimxfLim
x
xLimxfLim
x
xLimxfLim
Solução
x
xLimxfLim
x
x
x
x
xx
xx
xx
xx
xx
xx
xx
xx
( ) ( )
( ) ( )
3
2)(
111
11)(
1
1)(
111
11)(
1
1)(
1
1)(
1
1)(
:
1
1)()13
1
21
211
2211
33
22
11
3
2
11
6
3 6
11
6
3
11
=
++
+
=
++
+
=
+⋅+⋅−
−⋅+
=
−
−
=
−
−
=
−
−
=
=→
−
−
=
→
→
→→
→→
→→
→→
→→
→→
tfLim
tfLim
tt
tLimtfLim
ttt
ttLimtfLim
t
tLimtfLim
t
tLimtfLim
t
tLimtfLim
txFaça
Solução
x
xLimxfLim
t
t
tt
tt
tt
tt
tt
xx