Cálculo I: Limites e Funções

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Cálculo I: Lista II
Lista II
1. Calcule os seguintes limites, especificando em
cada passagem a propriedade ou teorema utilizado.
(a) lim
x→2
(3x2 − 5x+ 2)
(b) lim
x→−1
x2 + 2x− 3
4x− 3
(c) lim
x→1
(
2x2 − x+ 1
3x− 2
)2
(d) lim
x→−2
3
√
x3 + 2x2 − 3x+ 2
x2 + 4x+ 3
(e) lim
x→1
(4x2 − 7x+ 5)
(f) lim
x→−1
(x3 − 2x2 − 4x+ 3)
(g) lim
x→2
3x+ 2
x2 − 6x+ 5
(h) lim
x→−1
3x2 − 5x+ 4
2x+ 1
(i) lim
x→−3
x2 + 2x− 3
5− 3x
(j) lim
x→2
(
3x2 − 2x− 5
−x2 + 3x+ 4
)3
(k) lim
x→4
(
x3 − 3x2 − 2x− 5
2x2 − 9x+ 2
)2
(l) lim
x→−1
√
2x2 + 3x− 4
5x− 4
(m) lim
x→−2
3
√
3x3 − 5x2 − x+ 2
4x+ 3
(n) lim
x→2
3
√
2x2 + 3x+ 2
6− 4x
2. Calcule os limites:
(a) lim
x→1
x2 − 1
x− 1
(b) lim
x→−2
4− x2
2 + x
(c) lim
x→ 3
2
4x2 − 9
2x− 3
(d) lim
x→3
x2 − 4x+ 3
x2 − x− 6
(e) lim
x→ 1
2
2x2 + 5x− 3
2x2 − 5x+ 2
(f) lim
x→− 3
2
6x2 + 11x+ 3
2x2 − 5x− 12
(g) lim
x→1
x3 − 1
x2 − 1
(h) lim
x→−2
8 + x3
4− x2
(i) lim
x→2
x4 − 16
8− x3
3. Seja a função f definida por:
f(x) =
x2 − 3x+ 2
x− 1
se x ̸= 1
3 se x = 1
Calcule lim
x→1
f(x)
4. Seja a função f definida por:
f(x) =
2x2 − 3x− 2
x− 2
se x ̸= 2
3 se x = 2
Calcule lim
x→2
f(x)
5. Seja a função f definida por:
f(x) =
2x2 + 9x+ 9
x+ 3
se x ̸= −3
3 se x = −3
Mostre que lim
x→−3
f(x) = −3
6. Calcule os limites:
(a) lim
x→−1
x3 + 3x2 − x− 3
x3 − x2 + 2
1
Cálculo I: Lista II
(b) lim
x→3
x3 − 6x− 9
x3 − 8x− 3
(c) lim
x→1
x3 − 3x2 + 6x− 4
x3 − 4x2 + 8x− 5
(d) lim
x→2
x4 − 10x+ 4
x3 − 2x2
7. Calcule os limites:
(a) lim
x→1
√
x− 1
x− 1
(b) lim
x→0
1−
√
1− x
x
(c) lim
x→1
√
x+ 3− 2
x− 1
(d) lim
x→0
√
1− 2x− x2 − 1
x
(e) lim
x→0
√
1 + x−
√
1− x
x
(f) lim
x→1
√
2x−
√
x+ 1
x− 1
(g) lim
x→1
3−
√
10− x
x2 − 1
(h) lim
x→3
2−
√
x+ 1
x2 − 9
(i) lim
x→1
√
x+ 3− 2
x2 − 3x+ 2
(j) lim
x→2
x2 − 4√
x+ 2−
√
3x− 2
(k) lim
x→1
√
x2 − 3x+ 3−
√
x2 + 3x− 3
x2 − 3x+ 2
(l) lim
x→4
√
2x+ 1− 3
√
x− 2−
√
2
(m) lim
x→6
4−
√
10 + x
2−
√
10− x
(n) lim
x→0
√
3x+ 4−
√
x+ 4√
x+ 1− 1
(o) lim
x→2
√
x2 + x− 2−
√
x2 − x+ 2√
x2 − 2
Nos exerćıcios (8) (17), é dada uma função f . Cal-
cule os limites indicados, se existirem; se o(s) li-
mite(s) não existir(em), especifique a razão.
8.
f(x) =

3x− 2 se x > 1
2 se x = 1
4x+ 1 se x < 1
(a) lim
x→1+
f(x)
(b) lim
x→1−
f(x)
(c) lim
x→1
f(x)
9.
f(x) =
{
3− 2x se x ≥ −1
4− x se x < −1
(a) lim
x→−1+
f(x)
(b) lim
x→−1−
f(x)
(c) lim
x→−1
f(x)
10.
f(x) =
{
2x− 5 se x ≥ 3
4− 5x se x < 3
(a) lim
x→3+
f(x)
(b) lim
x→3−
f(x)
(c) lim
x→3
f(x)
11.
f(x) =

1− x2 se x < 2
0 se x = 2
x− 1 se x > 2
(a) lim
x→2+
f(x)
(b) lim
x→2−
f(x)
(c) lim
x→2
f(x)
12.
f(x) =
{
x2 − 3x+ 2 se x ≤ 3
8− 2x se x > 3
(a) lim
x→3+
f(x)
2
Cálculo I: Lista II
(b) lim
x→3−
f(x)
(c) lim
x→3
f(x)
13.
f(x) =

2x2 − 3x+ 1 se x < 2
1 se x = 2
−x2 + 6x− 7 se x > 2
(a) lim
x→2+
f(x)
(b) lim
x→2−
f(x)
(c) lim
x→2
f(x)
14. f(x) =
|x+ 1|
x+ 1
(a) lim
x→−1+
f(x)
(b) lim
x→−1−
f(x)
(c) lim
x→−1
f(x)
15. f(x) =
|3x− 2|
2− 3x
(a) lim
x→ 2
3
+
f(x)
(b) lim
x→ 2
3
−
f(x)
(c) lim
x→ 2
3
f(x)
16. f(x) =
x2 − 5x+ 4
|x− 1|
(a) lim
x→1+
f(x)
(b) lim
x→1−
f(x)
(c) lim
x→1
f(x)
17. f(x) =
|3x2 − 5x− 2|
x− 2
(a) lim
x→2+
f(x)
(b) lim
x→2−
f(x)
(c) lim
x→2
f(x)
18. Dada a função f definida por
f(x) =

3x− 2 se x > −1
3 se x = −1
5− ax se x < −1
Determine a ∈ R para que exista lim
x→−1
f(x).
19. Dada a função f definida por
f(x) =
{
4x+ 3 se x ≤ −2
3x+ a se x > −2
Determine a ∈ R para que exista lim
x→−2
f(x).
20. Dada a função f definida por
f(x) =
3x2 − 5x− 2
x− 2
se x < 2
3− ax− x2 se x ≥ 2
Determine a ∈ R para que exista lim
x→2
f(x).
21. Calcule:
(a) lim
x→1
3x+ 2
(x− 1)2
(b) lim
x→2
1− x
(x− 2)2
(c) lim
x→1
3x− 4
(x− 2)2
(d) lim
x→1
2x+ 3
(x− 1)2
(e) lim
x→1
1− 3x
(x− 1)2
(f) lim
x→0
3x2 − 5x+ 2
x2
(g) lim
x→−1
5x+ 2
|x+ 1|
(h) lim
x→−2
2x2 + 5x− 3
|x+ 2|
22. Calcule os limites laterais.
3
Cálculo I: Lista II
(a) lim
x→−2−
x+ 4
x+ 2
(b) lim
x→−2+
x+ 4
x+ 2
(c) lim
x→3−
1− 2x
x− 3
(d) lim
x→3+
1− 2x
x− 3
(e) lim
x→ 5
2
−
3x+ 2
5− 2x
(f) lim
x→ 5
2
+
3x+ 2
5− 2x
(g) lim
x→1−
2x+ 3
(x− 1)3
(h) lim
x→1+
2x+ 3
(x− 1)3
(i) lim
x→2−
2x2 − 3x− 5
(2− x)3
(j) lim
x→2+
2x2 − 3x− 5
(2− x)3
23. Calcule os limites no infinito.
(a) lim
x→+∞
(4x2 − 7x+ 3)
(b) lim
x→+∞
(−3x3 + 2x2 − 5x+ 3)
(c) lim
x→−∞
(5x3 − 4x2 − 3x+ 2)
(d) lim
x→−∞
(3x4 − 7x3 + 2x2 − 5x− 4)
(e) lim
x→+∞
(2x+ 3)
(f) lim
x→−∞
(4− 5x)
(g) lim
x→+∞
(5x2 − 4x+ 3)
(h) lim
x→+∞
(4− x2)
(i) lim
x→−∞
(3x3 − 4)
(j) lim
x→−∞
(8− x3)
(k) lim
x→+∞
3x+ 2
5x− 1
(l) lim
x→−∞
5− 4x
2x− 3
(m) lim
x→+∞
5x2 − 4x+ 3
3x+ 2
(n) lim
x→−∞
4x− 1
3x2 + 5x− 2
(o) lim
x→+∞
3− 2x
5x+ 1
(p) lim
x→−∞
4x− 3
3x+ 2
(q) lim
x→+∞
x2 − 4
x+ 1
(r) lim
x→−∞
x3 − 1
x2 + 1
(s) lim
x→+∞
x2 − 3x+ 4
3x3 + 5x2 − 6x+ 2
(t) lim
x→−∞
x2 + 4
8x3 − 1
(u) lim
x→−∞
x2 + x+ 1
(x+ 1)3 − x3
(v) lim
x→+∞
(2x− 3)3
x(x+ 1)(x+ 2)
(w) lim
x→−∞
(3x+ 2)3
2x(3x+ 1)(4x− 1)
(x) lim
x→+∞
(2x− 3)3(3x− 2)2
x5
(y) lim
x→−∞
(x+ 2)4 − (x− 1)4
(2x+ 3)3
Respostas
4
Cálculo I: Lista II
1.
(a) 4
(b) 4
7
(c) 4
(d) −2
(e) 2
(f) 4
(g) −8
3
(h) −12
(i) 0
(j) 1
8
(k) 9
4
(l)
√
5
3
(m) 2
(n) −2
2.
(a) 2
(b) 4
(c) 6
(d) 2
5
(e) −7
3
(f) 7
11
(g) 3
2
(h) 3
(i) −8
3
3. −1
4. 5
5.
6.
(a) −4
5
(b) 21
19
(c) 1
(d) 11
2
7.
(a) 1
2
(b) 1
2
(c) 1
4
(d) −1
(e) 1
(f)
√
2
4
(g) 1
12
(h) − 1
24
(i) −1
4
(j) −8
(k) 3
(l) 2
√
2
3
(m) −1
2
(n) 1
(o) 2
8.
(a) 1
(b) 5
(c) Não existe
9.
(a) 5
(b) 5
(c) 5
10.
(a) 1
(b) −11
(c) Não existe
11.
(a) 1
(b) −3
(c) Não existe
12.
(a) 2
(b) 2
(c) 2
13.
(a) 1
(b) 1
(c) 1
14.
(a) 1
(b) −1
(c) Não existe
15.
(a) −1
(b) 1
(c) Não existe
16.
(a) −3
(b) 3
(c) Não existe
17.
(a) 7
(b) −7
(c) Não existe
18. a = −10
19. a = 1
20. a = −4
21.
(a) +∞
(b) −∞
(c) +∞
(d) +∞
(e) −∞
(f) +∞
(g) −∞
(h) −∞
5
Cálculo I: Lista II
22.
(a) −∞
(b) +∞
(c) +∞
(d) −∞
(e) +∞
(f) −∞
(g) −∞
(h) +∞
(i) −∞
(j) +∞
23.
(a) +∞
(b) −∞
(c) −∞
(d) +∞
(e) +∞
(f) +∞
(g) +∞
(h) −∞
(i) −∞
(j) +∞
(k) 3
5
(l) −2
(m) +∞
(n) 0
(o) −2
5
(p) 4
3
(q) +∞
(r) −∞
(s) 0
(t) 0
(u) 1
3
(v) 8
(w) 9
8
(x) 72
(y) 3
2
6
	Lista II
	Respostas