Lista de Exercícios - LIMITES FURG

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Universidade Federal do Rio Grande 
Cálculo I 
Professor: Cristiano Rodrigues Garibotti 
 
Cálculo de limites 
Parte I 
1. lim
ℎ→0
√3ℎ+1−1
ℎ
 
2. lim
𝑥→5
 
𝑥−5
𝑥2−25
 
3. lim
𝑥→−5
𝑥2+3𝑥−10
𝑥+5
 
4. lim
𝑡→1
𝑡2+𝑡−2
𝑡2−1
 
5. lim
𝑥→−2
−2𝑥−4
𝑥3+2𝑥²
 
6. lim
𝑢→1
𝑢4−1
𝑢3−1
 
7. lim
𝑥→9
√𝑥−3
𝑥−9
 
8. lim
𝑥→1
𝑥−1
√𝑥+3−2
 
9. lim
𝑥→2
√𝑥²+12−4
𝑥−2
 
10. lim
𝑥→−3
2−√𝑥2−5
𝑥+3
 
11. Suponha lim
𝑥→𝑐
𝑓(𝑥) = 5 e 
lim
𝑥→𝑐
𝑔(𝑥) = −2. Determine: 
a)lim
𝑥→𝑐
𝑓(𝑥)𝑔(𝑥) 
b)lim
𝑥→𝑐
2𝑓(𝑥)𝑔(𝑥) 
c)lim
𝑥→𝑐
(𝑓(𝑥) + 3𝑔(𝑥)) 
d)lim
𝑥→𝑐
𝑓(𝑥)
𝑓(𝑥)−𝑔(𝑥)
 
12. Suponha lim
𝑥→𝑏
𝑓(𝑥) = 7 e 
lim
𝑥→𝑏
𝑔(𝑥) = −3. Determine: 
a)lim
𝑥→𝑏
(𝑓(𝑥) + 𝑔(𝑥)) 
b)lim
𝑥→𝑏
𝑓(𝑥). 𝑔(𝑥) 
c)lim
𝑥→𝑏
4𝑔(𝑥) 
d)lim
𝑥→𝑏
𝑓(𝑥)
𝑔(𝑥)
 
 
Usando o teorema do 
confronto 
 
13. Se √5 − 2𝑥² ≤ 𝑓(𝑥) ≤ 
√5 − 2𝑥² para −1 ≤ 𝑥 ≤ 1, 
determine lim
𝑥→0
𝑓(𝑥). 
 
Teoria e exemplos 
14. Se lim
𝑥→4
𝑓(𝑥)−5
𝑥−2
= 1, 
determine lim
𝑥→4
𝑓(𝑥). 
15. a) se lim
𝑥→2
𝑓(𝑥)−5
𝑥−2
= 3, 
determine lim
𝑥→2
𝑓(𝑥). 
 b) se lim
𝑥→2
𝑓(𝑥)−5
𝑥−2
= 4, 
determine lim
𝑥→2
𝑓(𝑥). 
 
Respostas: 
1.3/2 2. 1/10 3. -7 4. 3/2 5. -1/2 6. 4/3 7. 1/6 8. 4 9. ½ 
10. 3/2 11. (a) -10 (b) -20 (c) -1 (d) 5/7 12. (a) 4 (b) -21 (c) -12 
(d)5/7 13. √5 14. 7 15. (a) 5 (b) 5 
 
Parte II 
Obtendo limites Laterais 
algebricamente 
16. lim
ℎ→0+
√ℎ2+4ℎ+5−√5
ℎ
 
17. (a) lim
𝑥→−2+
(𝑥 + 3)
|𝑥+2|
𝑥+2
 
(b) lim
𝑥→−2−
(𝑥 + 3)
|𝑥+2|
𝑥+2
 
 
Usando lim
𝜃→0
sen 𝜃
𝜃
= 1 
Determine os limites nos 
exercícios: 
18. lim
𝜃→0
𝑠𝑒𝑛√2𝛳
√2𝛳
 
19. lim
𝑦→0
𝑠𝑒𝑛 (3𝑦)
4𝑦
 
20. lim
𝑥→0
𝑡𝑔(2𝑥)
𝑥
 
21. lim
𝑥→0
𝑥𝑐𝑜𝑠𝑒𝑐(2𝑥)
cos (5𝑥)
 
22. lim
𝑥→0
𝑥+cos(𝑥)
𝑠𝑒𝑛(𝑥)cos (𝑥)
 
23. lim
𝜃→0
𝑠𝑒𝑛(𝜃)
𝑠𝑒𝑛(2𝜃)
 
24. lim
𝑥→0
𝑡𝑔(3𝑥)
𝑠𝑒𝑛(8𝑥)
 
 
Calculando limites quando 
x→ ±∞ 
Nos exercícios 25-36, encontre o 
limite de cada função quando (a) x 
→ ∞ e x→ -∞. 
25. 𝑓(𝑥) = 
2
𝑥
− 3 
26. 𝑔(𝑥) = 
1
2+(
1
𝑥
)
 
27. ℎ(𝑥)
−5+( 
7
𝑥
 )
3−(
1
 𝑥2
 )
 
Encontre os limites: 
28. lim
𝑥→∞
 
𝑠𝑒𝑛(2𝑥)
𝑥
 
29. lim
𝑡→−∞
2−𝑡+𝑠𝑒𝑛(𝑡)
𝑡+cos (𝑡)
 
30. lim
𝑥→∞
𝑒−𝑥𝑠𝑒𝑛(𝑥) 
31. lim
𝑥→−∞
𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥
 
 Limites de funções racionais 
32.𝑓(𝑥) = 
2𝑥+3
5𝑥+7
 
33.𝑓(𝑥) =
𝑥+1
𝑥2+3
 
34.ℎ(𝑥) = 
7𝑥³
𝑥3−3𝑥2+6𝑥
 
35.𝑔(𝑥) = 
10𝑥5+𝑥4+31
𝑥6
 
36.ℎ(𝑥) = 
−2𝑥3−2𝑥+3
3𝑥3+3𝑥2−5𝑥
 
 
Limites infinitos 
 
Determine os limites nos 
exercícios: 
37. lim
𝑥→0+
1
3𝑥
 
38. lim
𝑥→2−
3
𝑥−2
 
39. lim
𝑥→−8+
2𝑥
𝑥+8
 
40. lim
𝑥→7
4
(𝑥−7)²
 
41.a) lim
𝑥→0+
2
3𝑥1/3
 
b) lim
𝑥→0−
2
3𝑥1/3
 
42.lim
𝑥→0
4
𝑥2/5
Respostas: 
16. 2/√5 17. (a) 1 (b) -1 18. 1 19. 3/4 20. 2 21. 1/2 22. 2 
23. 1 /2 24. 3/8 25. (a) -3 (b) -3 26. (a)1/2 (b)1/2 27. (a) -5/3 
(b) -5/3 28. 0 29. -1 30. 0 31. -1 32. (a) 2/5 (b) 2/5 33. (a) 0 (b) 
0 34. (a)7 (b) 7 35. (a) 0 (b) 0 36. (a) -2/3 (b) -2/3 37.∞ 38. -∞ 
39. -∞ 40. ∞ 41. (a) ∞ (b) -∞ 42. ∞