Exercícios de Cálculo Diferencial e Integral I

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Universidade Veiga de Almeida
Curso: Ba´sico das engenharias
Disciplina: Ca´lculo Diferencial e Integral I
Professora: Adriana Nogueira
2a Lista de Exerc´ıcios
Exerc´ıcio 1: Calcule os limites dados abaixo:
(a) lim
x→+∞
2x2 − 5
3x4 + x+ 2
(b) lim
x→+∞
2x3 − 5
3x2 + x+ 2
(c) lim
x→+∞
5x2 − 3x+ 1
2x2 + 4x− 7
(d) lim
x→−∞
4− 7x
2 + 3x
(e) lim
x→−∞
2x2 − 3
4x3 + 5x
(f) lim
x→+∞
2− x2
x+ 3
(g) lim
x→+∞
3
√
x+ 1
x+ 1
(h) lim
x→+∞
x2
10 + x
√
x
(i) lim
x→−∞
6x2
3
√
5x6 − 1
Exerc´ıcio 2: Calcule os limites dados abaixo:
(a) lim
x→7
√
x−√7√
x+ 7−√14 (b) limx→1
1
x2 − 1
x− 1 (c) limx→p
1
x − 1p
x− p
(d) lim
x→2+
√
x2 − 4 +√x− 2√
x− 2 (e) limx→3+
4x2
9− x2 (f) limx→3−
4x2
9− x2
(g) lim
x→0+
√
3 + x2
x
(h) lim
x→0−
√
3 + x2
x
(i) lim
x→3+
√
x2 − 9
x− 3
(j) lim
x→4+
x
x− 4 (k) limx→4−
x
x− 4 (l) limx→2+
x+ 2
x2 − 4
(m) lim
x→2−
x+ 2
x2 − 4 (n) limx→5+
4x
(x− 5)2 (o) limx→5−
4x
(x− 5)2
(p) lim
x→1+
x− 3
x2 − 1 (q) limx→1−
4− x2
x3 − 1 (r) limx→3−
2x2 + 5x+ 1
x2 − x− 6
(s) lim
x→+∞
√
3 + x2
x
(t) lim
x→−∞
√
3 + x2
x
1
Exerc´ıcio 3: Calcule os valores de a e b para que tenhamos a identidade
abaixo:
lim
x→+∞[ax+ b−
x3 + 1
x2 + 1
] = 0.
Exerc´ıcio 4: Determine, caso existam, as ass´ıntotas horizontais e verticais
dos gra´ficos das func¸o˜es dadas abaixo:
(a) f(x) =
x2 − 5
2x2 − 4 (b) f(x) =
x3 + 5x2 + 1
x2 + 1
(c) f(x) =
x2 + 5x+ 4
x5 + 1
RESPOSTAS:
1) (a) 0 (b) +∞ (c) 5/2 (d) -7/3 (e) 0 (f) −∞
(g) 0 (h)+∞ (i) 6
3
√
5
2) (a)
√
2 (b) −2 (c) −1
p2
(d) 3 (e)−∞ (f) +∞ (g) +∞
(h) −∞ (i) +∞ (j) +∞ (k) −∞ (l) +∞ (m) −∞ (n) +∞
(o) +∞ (p) −∞ (q) −∞ (r) −∞ (s) 1 (t) −1
3) a = 1 e b = 0
4) a) y = 1/2 e´ ass´ıntota horizontal e x =
√
2 e x = −√2 sa˜o ass´ıntotas
verticais.
b) Na˜o ha´ ass´ıntotas.
c) y = 0 e´ ass´ıntota horizontal e na˜o ha´ ass´ıntota vertical.
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