Introdução ao Cálculo Diferencial e Integral - Funçoes
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Introdução ao Cálculo Diferencial e Integral - Funçoes


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Respostas 
1) 8 
2) 4 
3) 526 \u2212\u2212 
4) -10 
5) -3 
6) -4 
7) 3
1\u2212 
8) 12 
9) 80 
10) 2 
11) 0 
12) 4 
13) 4 
14) 4
1\u2212 
15) 2 
16) 3
4 
17) 14
5 
 
 
 
 
 
 
 
Cálculo Diferencial e Integral 
 55
AULA 08 
 
 
6.2 - LIMITES INFINITOS: 
 Quando os valores assumidos pela variável x são tais que |x|> N, sendo N tão grande 
quanto se queria, então se diz que o limite da variável x é infinito. 
 
+\u221e=+\u221e\u2192 xxlim ou \u2212\u221e=\u2212\u221e\u2192 xxlim 
 
 
 
6.2.1 - Igualdades Simbólicas: 
 
6.2.1.1 \u2013 Tipo Soma: 
a. (3) + ( \u221e± ) = \u221e± 
b. (+\u221e ) + (+\u221e ) = + \u221e 
c. - \u221e + (-\u221e ) = - \u221e 
d. \u221e - \u221e = indeterminado 
 
6.2.1.2 \u2013 Tipo Produto: 
a. 5 x ( \u221e± ) = \u221e± 
b. (-5) x ( \u221e± ) = \u221em 
c. (+\u221e )x(+\u221e ) = + \u221e 
d. (+\u221e )x(-\u221e ) = -\u221e 
e. ± \u221e x 0 = indeterminado 
 
6.2.1.3 \u2013 Tipo Quociente: 
a. 0=\u221e
c
 
b. \u221e=\u221e
c
 
c. 00 =\u221e 
d.
0
0
 e =\u221e
\u221e
 indeterminado 
 
6.2.1.4 \u2013 Tipo Potência: 
a. +\u221e=+\u221ec (c>1) 
b. 0=+\u221ec (0<c<1) 
c. 00 =\u221e 
d. 0=\u2212\u221ec 
e. +\u221e=+\u221e +\u221e)( 
f. \u2212\u221e=\u2212\u221e c)( (se c for ímpar) 
g. +\u221e=\u2212\u221e c)( (se c for par) 
h. 0)( =+\u221e \u2212\u221e 
i. 0)( =±\u221e \u2212c 
j. 00 = indeterminado 
k. =±\u221e 0)( indeterminado 
l. =±\u221e1 indetermindado 
Cálculo Diferencial e Integral 
 56
 
 
Obs.: O limite de uma função polinomial quando x tende ao infinito, é o limite do termo de maior 
grau. 
 
Exemplos: 
 
1) =\u2212++\u221e\u2190 )13(lim 2 xxx 
 
 
 
2) =\u2212+
\u2212+\u2212
+\u221e\u2192 432
1245lim 2
2
xx
xxx
x 
 
 
 
3) =+\u2212
\u2212+
\u2212\u221e\u2192 3
543lim 2
2
xx
xx
x 
 
 
 
4) \u2212\u221e\u2192xlim =+3 4
5
6
2
x
x
 
 
 
 
 
5) =\u2212+
+
+\u221e\u2192 132
18lim 4
4
xx
xx
x 
 
 
 
 
6) =\u2212\u2212\u2212+++\u221e\u2192 )11(lim 22 xxxxx 
Cálculo Diferencial e Integral 
 57
AULA 08\u2013 EXERCÍCIOS 
 
1) =\u2212\u2212\u2212+\u221e\u2192 )1235(lim 23 xxxx 
2) =\u2212+\u2212\u2212\u221e\u2192 )122(lim 245 xxxx 
3) =\u2212+\u2212\u2212\u221e\u2192 )123(lim 24 xxx 
4) =+++\u221e\u2192 )853(lim 24 xxx 
5) =\u2212+\u2212\u2212\u221e\u2192 )235(lim 3 xxx 
6) =\u2212+\u2212+\u221e\u2192 )23(lim 2 xxx 
7) =\u2212+
\u2212+\u2212
+\u221e\u2192 3
132lim 2
23
xx
xxx
x 
8) =\u2212
+
\u2212\u221e\u2192 1
12lim 2
2
x
x
x 
9) =\u2212\u2212\u221e\u2192 3
3lim 2x
x
x 
10) =\u2212+\u2212
++\u2212
\u2212\u221e\u2192 359
1253lim 23
23
xxx
xxx
x 
11) =+\u2212
\u2212+
\u2212\u221e\u2192 784
852lim 5
23
xx
xx
x 
12) =+
+\u2212
\u2212\u221e\u2192 7
125lim
23
x
xx
x 
13) =\u2212+
++
\u2212\u221e\u2192 33
2
)1(
1lim
xx
xx
x 
14) =+
++
+\u221e\u2192 1
1lim
2
x
xx
x 
15) =+
++
\u2212\u221e\u2192 1
1lim
2
x
xx
x 
16) =
+
\u2212\u2212
+\u221e\u2192
1
532lim
4
2
x
xx
x 
17) =
+
\u2212\u2212
\u2212\u221e\u2192
1
532lim
4
2
x
xx
x 
18) =\u2212+++\u221e\u2192 )43(lim 2 xxxx 
19) =\u2212++\u2212\u221e\u2192 )43(lim 2 xxxx 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Respostas: 
1) + \u221e 
2) - \u221e 
3) - \u221e 
4) +\u221e 
5) +\u221e 
6) -\u221e 
7) +\u221e 
8) 2 
9) 0 
10) 3
1 
11) 0 
12) +\u221e 
13) 3
1 
14) 1 
15) -1 
16) 2 
17) 2 
18) 2
3 
19) +\u221e 
 
 
 
 
 
 
Cálculo Diferencial e Integral 
 58
AULA 09 
 
 
6.3 \u2013 LIMITES TRIGONOMÉTRICOS: 
 1lim 0 =\u2192 x
senx
x 
 
Demonstrando o limite fundamental por tabela temos que: 
 
Usando valores de x\u2192 0 em radianos, obtemos valores 
iguais ou muito próximos. 
 
 
 
 
 
Exemplos: 
 
1) =\u2192 x
xsen
x
3lim 0 
 
 
2) =\u2212\u2192 20 cos1lim x
x
x 
 
 
 
3) =\u2192 xsen
xsen
x 2
5lim 0 
 
 
 
4) =+
+
\u2192 xsenxsen
senxxsen
x 42
5lim 0 
 
 
 
 
5) =+
+
\u2192 xsenx
xsenx
x 9
23lim 0 
 
 
 
 
 
x Senx 
0,008 0,008 
0,006 0,006 
0,004 0,004 
0,002 0,002 
0,001 0,001 
Cálculo Diferencial e Integral 
 59
6) =\u2192 x
tgx
x 0lim 
 
 
 
 
 
 
 
7) =\u2212\u2192 x
x
x
cos1lim 0 
 
 
 
 
 
 
 
8) =\u2192 sennx
senmx
x 0lim 
 
 
 
 
 
 
 
AULA 09 \u2013 EXERCÍCIOS 
 
1) =\u2192 x
xsen
x 2
3lim 0 
2) =\u2192 x
senx
x 4
lim 0 
3) =\u2192 x
xtg
x 3
2lim 0 
4) =\u2192 xsen
xsen
x 3
4lim 0 
5) =\u2192 xtg
xtg
x 5
3lim 0 
6) =\u2212\u2192 xsenx
x
x
cos1lim 0 
7) =\u2212\u2192 20 sec1lim x
x
x 
8) =+\u2192 x
senxtgx
x 0lim 
9) =\u2212
\u2212
\u2192 tgx
xsenx
x 1
coslim 0 
10) =\u2212\u2192 xsen
senxtgx
x 20lim 
11) =+
\u2212
\u2192 senxx
senxx
x 0lim 
12) =\u2212\u2192 xsen
xx
x 4
3cos5coslim 0 
13) =\u2212\u2192 senx
xsenxsen
x
23lim 0 
14) =\u2212+\u2192 x
senaaxsen
x
)(lim 0 
15) =\u2212\u2192 20 3
2cos1lim
x
x
x 
 
Respostas: 
1) 3/2 
2) ¼ 
3) 2/3 
4) 4/3 
5) 3/5 
6) ½ 
7) \u2013 ½ 
8) 2 
9) -1 
10) 0 
11) 0 
12) 0 
13) 1 
14) cos a 
15) 2/3 
Cálculo Diferencial e Integral 
 60
AULA 10 
 
6.4 \u2013 LIMITES DE FUNÇÃO EXPONENCIAL E LOGARÍTMICAS: 
 
 ex
x
x =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u221e\u2192 11lim (1) 
 
 
 
 Neste caso, e representa a base dos logaritmos naturais ou neperianos. Trata-se do 
número irracional e, cujo valor aproximado é 2,7182818 
 
 
 Nota-se que a medida que x \u221e\u2192 , 
x
x
\u239f\u23a0
\u239e\u239c\u239d
\u239b + 11 \u2192 e 
 
 De forma análoga, efetuando a substituição y
x
=1 e 
y
x 1= 
temos: 
ey yy =+\u2192
1
0 )1(lim (2) 
 
 Ainda de forma mais geral, temos: 
 
(3) kly
l
y eky =+\u2192 )1(lim 0 
 
 
(4) kl
lx
x ex
k =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u221e\u2192 1lim 
 
 
(5) a
x
a x
x ln
1lim 0 =\u2212\u2192 
 
 
(6) 1
1lim 0 =\u2212\u2192 x
e x
x 
 
Exemplos: 
1) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u221e\u2192
x
x x
431lim 
 
 
 
2) =+\u2192 xx x
3
0 )21(lim 
 
X 
x
x
\u239f\u23a0
\u239e\u239c\u239d
\u239b + 11 
1 2 
2 2,25 
3 2,3703 
10 2,5937 
100 2,7048 
1000 2,7169 
10000 2,7181 
100000 2,7182 
Cálculo Diferencial e Integral 
 61
 
3) =\u2212\u2192 x
x
x 2
13lim 0 
 
 
 
4) =\u2212\u2192 xsen
e x
x 2
1lim 0 
 
 
 
5) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u221e\u2192
x
x x
251lim 
 
 
 
6) ( ) =+\u2192 xx x 20 21lim 
 
 
 
7) =\u2212\u2192 x
x
x
12lim 0 
 
 
 
8) =\u2212\u2192 1
3lim 0 xx e
xsen
 
 
 
 
 
9) =\u2212\u2192 xsen
e x
x 4
1lim
3
0 
 
 
 
10) =\u2212\u2192 xsen
x
x 2
13lim
5
0 
 
 
 
 
 
11) =\u2212+
\u2212\u2212
\u2212\u2192 26
413loglim 2 x
x
x
Cálculo Diferencial e Integral 
 62
AULA 10 \u2013 EXERCÍCIOS 
 
1) =\u2212
\u2212
\u2192 2
4
2
2
3lim x
x
x 
2) =\u2212
\u2212
\u2192
1
1
1lim x
x
x e 
3) =\u239f\u23a0
\u239e\u239c\u239d
\u239b \u2212
+\u2212
\u2192
2
45
4
2
1lim
x
xx
x e
 
4) =++
++
\u2212\u2192 45
23loglim 2
2
31 xx
xx
x 
5) =\u2212+
\u2212
\u2192 21
3lnlim 3 x
x
x 
6) =+
\u2212
\u2192 xx
xx
x 2
3
0 loglim 
7) =\u239f\u23a0
\u239e\u239c\u239d
\u239b ++\u221e\u2192
x
x x
211lim 
8) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u2212\u221e\u2192
311lim
x
x x
 
9) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +
+
+\u221e\u2192
211lim
x
x x
 
10) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +
\u2212
+\u221e\u2192
311lim
x
x x
 
11) =\u239f\u23a0
\u239e\u239c\u239d
\u239b +\u2212\u221e\u2192
x
x x
41lim 
12) =\u239f\u23a0
\u239e\u239c\u239d
\u239b ++\u221e\u2192
x
x x
321lim 
13) =\u239f\u23a0
\u239e\u239c\u239d
\u239b \u2212\u2212\u221e\u2192
x
x x
321lim 
14) =+\u2192 xx x
1
0 )41(lim 
15) =\u2212\u2192 xx x
2
0 )31(lim 
16) =\u239f\u23a0
\u239e\u239c\u239d
\u239b
\u2212
\u2212 +
+\u221e\u2192
3
1
4lim
x
x x
x
 
17) =\u239f\u239f\u23a0
\u239e
\u239c\u239c\u239d
\u239b
\u2212
+
+\u221e\u2192
2
3
1lim 2
2 x
x x
x
 
18) =\u239f\u23a0
\u239e\u239c\u239d
\u239b
+
+
+\u221e\u2192
x
x x
x
12
32lim 
19) =+\u2192 x
x
x 2
)1ln(lim 0 
20) =+\u2192 x
x
x 3
)21ln(lim 0 
Respostas 
 
1) 81 
2) e2 
3) e-12 
4) -1 
5) ln4 
6) 0 
7) e2 
8) e1/3 
9)
Anderson
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Excelente material, obrigado pela ajuda
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