limites

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1
Ca´lculo I - Exerc´ıcios
1 Limites
Questa˜o 1. Considere a func¸a˜o f(x) dada pelo gra´fico abaixo:
Intuitivamente, encontre, se existir:
1. lim
x→ 1+
f(x)
2. lim
x→ 1−
f(x)
3. lim
x→ 1
f(x)
4. lim
x→ +∞
f(x)
5. lim
x→ −∞
f(x)
Questa˜o 2. Calcule os seguintes limites, caso existam:
1. lim
x→ 0
|x|
x
2. lim
x→ 0
|x|
x2
3. lim
x→ 0
|x− 1|
1− x
4. lim
x→ 3
1
|x− 3|
5. lim
x→ −1
x3 − 4x2 − 6x− 1
x2 + x
6. lim
x→ 0
x4 − 3x3 + 7x2 − x
x
7. lim
x→ 5
√
x−√5
x− 5
8. lim
x→ 3
x2 − 9√
3−√x
9. lim
x→ 1
3
√
x+ 7− 2
x− 1
10. lim
x→ 4
3
√
x− 1
4
√
x− 1
11. lim
x→ 4
3−√5 + x
1−√5− x
12. lim
x→ 1
3
√
x2 − 2 3√x+ 1
(x− 1)2
13. lim
x→ 0
√
1 + x−√1− x
x
14. lim
x→ 0
ax − 1
x
, a > 0, a 6= 1
2
Questa˜o 3. Calcule os limites infinitos, caso existam:
1. lim
x→ +∞
x2 − 9√
3−√x
2. lim
x→ +∞
x
√
x+ 3x− 10
x3
3. lim
x→ +∞
√
3−√x
x2 − 9
4. lim
x→ +∞
x
(√
x2 − 1− x
)
5. lim
x→ +∞
10x2 − 3x+ 4
3x2 − 1
6. lim
x→ +∞
5x3 − x2 + x− 1
x4 + x3 − x+ 1
7. lim
x→ +∞
x3 − x√
5− 4x2
8. lim
x→ +∞
x2 − 2x+ 3
2x2 + 5x− 3
9. lim
x→ +∞
x+ 1
x2 + 1
10. lim
x→ −∞
−5x3 + 2
7x3 + 3
11. lim
x→ −∞
√
x2 + 1
x+ 1
12. lim
x→ −∞
(√
x2 + 1−
√
x2 − 1
)
13. lim
x→ −∞
3
√
3x7 − 4x5
2x7 + 1
14. lim
x→ +∞
8− x√
x2 − 7
Questa˜o 4. Utilizando os limites fundamentais lim
x→0
sinx
x
= 1, calcule os limites abaixo:
1. lim
x→ 0
tanx
x
2. lim
x→ 0
tan ax
x
, a 6= 0
3. lim
x→ 0
sin ax
sin bx
, b 6= 0
4. lim
x→ 0
cos 2x− cos 3x
x2
5. lim
x→ 0
sin 10x
sin 7x
6. lim
x→ 0
1− cosx
x
7. lim
x→ 0
sin 4x
3x
8. lim
x→ 0
sin3 (x/2)
x3
9. lim
x→ 0
cotx
x
10. lim
x→ 0
tan3
(
x+ 1
4
)
(x+ 1)3
3
Soluc¸a˜o 1.
1.
1
2
2. +∞
3. Na˜o existe
4.
1
2
5. −∞
Soluc¸a˜o 2.
1. Na˜o existe
2. ∞
3. Na˜o existe
4. ∞
5. −5
6. −1
7.
2
√
5
5
8. 12
√
3
9.
1
4
10.
4
3
11. −1
3
12.
1
27
13.
√
2
14. ln a
Soluc¸a˜o 3.
1. +∞
2. 0
3. 0
4. −1
2
5.
10
3
6. 0
7. +∞
8.
1
2
9. 1
10. −5
7
11. 1
12. 0
13. 3
√
3
2
14. −1
Soluc¸a˜o 4.
1. 1
2. a
3.
a
b
4.
5
2
5.
10
7
6. 1
7.
4
3
8.
1
8
9. +∞
10.
1
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