Calculus/Limits/Exercises

Proofs of Some Basic Limit Rules Differentiation Basic Limit Exercises 1. 2. 3. 4. Solutions One-Sided Limits Evaluate the following limits or state that the limit does not exist. 5. 6. 7. 8. The limit does not exist. 9. 10. Solutions Two-Sided Limits Evaluate the following limits or state that the limit does not exist. 11. 12. The limit does not exist. 13. The limit does not exist. 14. 15. 16. 17. 18. 19. 20. The limit does not exist. 21. 22. The limit does not exist. 23. 24. 25. The limit does not exist. 26. 27. 28. 29. 30. 31. The limit does not exist. 32. The limit does not exist. 33. The limit does not exist. Solutions Limits to Infinity Evaluate the following limits or state that the limit does not exist. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. Solutions Limits of Piecewise Functions Evaluate the following limits or state that the limit does not exist. 48. Consider the function a. b. c. The limit does not exist 49. Consider the function a. b. c. d. e. f. 50. Consider the function a. b. c. d. Solutions Intermediate Value Theorem 51. Use the intermediate value theorem to show that there exists a value for from . If you cannot use the intermediate value theorem to show this, explain why. Notice is continuous from . Ergo, the intermediate value theorem applies. For all , there exists a so that . 52. Use the intermediate value theorem to show that there exists an so that for from . If you cannot use the intermediate value theorem to show this, explain why. Notice is continuous from . Ergo, the intermediate value theorem applies. It is known the following is true: . From there, we can directly argue the following: By the intermediate value theorem, if is continuous from , then there exists an so that for . 53. Use the intermediate value theorem to show that there exists a value so that for from . If you cannot use the intermediate value theorem to show this, explain why. Notice is not continuous for since is unbounded. Ergo, the intermediate value theorem cannot be used to solve this problem. Solutions External Links Limits 1 exercise at Khan Academy Limits 2 exercise at Khan Academy Proofs of Some Basic Limit Rules Differentiation