Lista de exercícios sobre limites

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Universidade Federal de Santa Catarina
Centro de Cieˆncias F´ısicas e Matema´ticas
Departamento de Matema´tica
MTM3101 - Ca´lculo 1
3a lista de exerc´ıcios (14/08/2017 a 18/08/2017)
1. Encontre as ass´ıntotas verticais da func¸a˜o f(x) =
x2 + 1
3x− 2x2 .
2. Calcule, quando existirem, justificando as respostas, os seguintes limites:
lim
x→0
sen(3x)
x
(a) lim
x→0
tg x
2x
(b)
lim
x→0
2x
sen(3x)
(c) lim
x→0
tg(2x)
sen(3x)
(d)
lim
x→0
1− cosx
x2
(e) lim
x→0
x2 cos
(
1
x
)
(f)
lim
x→1
1− x2
sen(pix)
(g) lim
x→pi
1− sen (x/2)
pi − x(h)
lim
x→0
x− senx
x+ senx
(i) lim
x→pi
4
senx− cosx
1− tg x(j)
lim
x→0
x5 + 2x3
tg x− senx(k) limx→0
√
1 + sen x−√1− senx
x
(l)
3. Considere a func¸a˜o:
f(x) =

sen(x− 1)
x− 1 , se x < 1
α , se x = 1
β(x2 − 1)
x− 1 , se 1 < x < 2
γ(x2 + 1)
2x
, se x > 2.
Encontre valores reais para α, β e γ tais que a func¸a˜o f seja cont´ınua nos pontos x0 = 1 e x0 = 2.
4. Calcule:
lim
x→0+
sen2 x
x4 − x3(a) limx→5−
e−x
(x− 5)3(b)
lim
x→3+
ln(x2 − 9)(c) lim
x→0+
arctan(lnx)(d)
lim
x→3+
x2 − 9
x2 − 6x+ 9(e) limx→∞
(√
x+
√
x−√x− 2
)
(f)
lim
x→0−
1
x
(g) lim
x→0
1
|x|(h)
lim
x→1
2− x
(x− 1)2(i) limx→2
x2 − x+ 6
x− 2(j)
1
5. Escreva as quatro definic¸o˜es intuitivas de limites infinitos para limites laterais.
6. Encontre o limite.
lim
x→∞
2x6 − 7x+ 3
4x6 + x+ 5
(a) lim
x→−∞
7x4 + 1
x5 + 6x+ 1
(b)
lim
x→∞
√
2x4 − 3x− 5
2 + x2 + 3x4
(c) lim
x→∞
√
x2 + 5
6x+ 1
(d)
lim
t→∞
t2 − 4
t+ 1
(e) lim
u→∞
4u4 + 5
(u2 − 2)(2u2 − 1)(f)
lim
x→∞
sen2 x
x2 + 1
(g) lim
x→−∞
(
x4 + x5
)
(h)
lim
x→−∞
3x5 − x2 + 7
2− x2(i) limy→−∞
3− y√
5 + 4y2
(j)
lim
x→∞
√
x+ 3
√
x
x2 + 7
(k) lim
x→∞
(√
x+ 2−√x+ 5
)
(l)
lim
x→∞
(x2 + 1)1/3
x+ 1
(m) lim
x→∞
(√
x2 + 3x+ 2− x
)
(n)
7. Esboce o gra´fico de uma func¸a˜o f que satisfac¸a todas as condic¸o˜es dadas:
lim
x→3+
f(x) = 4, lim
x→3−
f(x) = 2, lim
x→−2
f(x) = 2, f(3) = 3 e f(−2) = 1.(a)
lim
x→2
f(x) = −∞, lim
x→∞
f(x) =∞, lim
x→−∞
f(x) = 0, lim
x→0+
f(x) =∞ e lim
x→0−
f(x) = 0.(b)
8. Calcule o limite abaixo, caso exista:
lim
x→0
sen
(
x2 + 1
x
)− sen ( 1
x
)
x
.
9. Escreva as definic¸o˜es precisas para
lim
x→−∞
f(x) =∞ e lim
x→∞
f(x) = −∞.
10. Calcule os seguintes limites:
lim
x→3pi/2
(1 + cos x)
1
cos x(a) lim
x→pi/2
(
1 +
1
tanx
)tanx
(b)
lim
n→∞
(
2n+ 3
2n+ 1
)n+1
(c) lim
x→∞
(
1 +
10
x
)x
(d)
11. Usando que lim
x→0
ax − 1
x
= ln a para a > 0, calcule:
lim
x→2
5x − 25
x− 2(a) limx→0
e−x − e−2x
x
(b)
lim
x→−3
4
x+3
5 − 1
x+ 3
(c) lim
x→2
10x−2 − 1
x− 2(d)
2