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2a. Lista de Exerc´ıcios de MAT 3110 BMAC - IMEUSP - 1o. sem. 2010 - Turma 54 Profa. Maria Izabel Ramalho Martins I. Limites e Continuidade

1. Calcule os seguintes limites, caso existam, justificando seu c´ alculo: 3 x→1 x − 2 −x3 − 2x2 + 4x + 8 4. lim x→−2 2x3 + 9x2 + 12x + 4 √ x2 + 16 − 5 7. lim x→−3 x2 + 3x sen(x2 − 3x + 2) 10. lim x→2 x−2 √ x4 + x2 13. lim x→0 √ x u2 + 12 − 4 √ 16. lim u→2 2 − u3 − 4 t+1 19. lim √ t→−1 5 t + 1 | x − 1| 22. lim− x−1 x→1 √ x3 + x2 − 5x + 3 25. lim x→1 x2 − 1 √ √ 28. lim 3 x + 1 − 3 x x→+∞ √ 5x6 + 7x4 + 7 31. lim x→+∞ x4 − 2 2x3 + 2 34. lim √ x→−∞ 4 7x12 + 5x4 + 7 1 37. lim x sen x→+∞ x 1.

lim

x−1 x→1,01 |x − 1| x−3 √ 5. 6. lim √ x→3 x− 3 x4 − x3 − x2 + 1 sen(21x) 8. lim 9. lim 2 x→1 x→0 sen(2010x) x +x−2 √ √ 4 5 2x − 1 x4 + 1 − 1 12. lim 11. lim + √ x→0 x4 x→1/2 2x − 1 2.

x−2 x→2 2x − 4 x2 + x − 56 lim 2 x→7 x − 11x + 28 lim

14. lim

x→0

17. lim

x→0

20. lim

x→2

23. lim t→0

26. lim

x→+∞

29. lim

x→−∞

32. lim

x→−∞

35. lim

u→2

38. lim

x→−∞

tg x x sen(sen(2x)) x √ 2 − x3 − 1 √ x2 + 3 − 2 √ t2 + 9 − 3 t sen t x √ x+1 √ 3 x+1 √ 3 4x + 1 3x5 + 2x2 − 4 √ x6 + x + 1 2 u2 − 3u + 2 cos x x

3.

lim

15. lim

x→0

18. limπ x→ 2

21. lim

x→2

24. lim− x→2

27. lim

x→−∞

30. lim

x→+∞

33. lim

x→−∞

1 − cos x x2 cos x x − π2 √ x2 + 12 − 4 √ 2 − x3 − 4 √ x2 − 4x + 4 x−2 2 − x + 5x2 − 5x5 2x3 − 2x2 + 5 √ √ x 2 + 1 − x4 + 1 √ x2 + 9 + x + 3

2 x2 − 3x + 2 x→1 (x2 − 2x) sen(x2 − 4) √ √ 39. lim x→2 x2 + 4 − 4x

36. lim+

x6 x6 2. Seja f : IR → IR tal que 1 + x2 + ≤ f (x) + 1 ≤ sec x2 + , para todo x ∈ IR. 3 3 Calcule, justificando, 1 a. lim f (x) ; b. lim f (x) cos . x→0 x→0 x + x2


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