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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. 2
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