SECTION 14.1 Investigating Limits Using Tables and Graphs 945 Skill Building In Problems 7–16, use a table to investigate the indicated limit. 7. ( ) → x lim 4 x 2 3 8. ( ) + → x lim 2 1 x 3 2 9. + + → x x lim 1 1 x 0 2 10. − + → x x lim 2 4 x 0 2 11. − − → x x x lim 4 4 x 4 2 12. − − → x x x lim 9 3 x 3 2 2 13. ( ) + → e lim 1 x x 0 14. − → − e e lim 2 x x x 0 15. − → x x lim cos 1 , x 0 x in radians 16. → x x lim tan , x 0 x in radians In Problems 17–22, use the graph shown to investigate the indicated limit. 17. ( ) → f x lim x 2 3 23 4 2 (2, 3) x y 18. ( ) → f x lim x 4 3 23 4 2 x y (4, 23) 19. ( ) → f x lim x 2 4 4 2 x y 2 (2, 2) 20. ( ) → f x lim x 2 2 4 4 1 3 x y 2 (2, 1) 21. ( ) → f x lim x 3 3 6 6 x y 3 (3, 6) 22. ( ) → f x lim x 4 4 8 8 2 6 x y 4 In Problems 23–42, graph each function. Use the graph to investigate the indicated limit. 23. ( ) ( ) = + → f x f x x lim , 3 1 x 4 24. ( ) ( ) = − →− f x f x x lim , 2 1 x 1 25. ( ) ( ) = − → f x f x x lim , 1 x 2 2 26. ( ) ( ) = − →− f x f x x lim , 1 x 1 3 27. ( ) ( ) = →− f x f x x lim , 2 x 3 28. ( ) ( ) = → f x f x x lim , 3 x 4 29. ( ) ( ) = π→ f x f x x lim , sin x 2 30. ( ) ( ) = π→ f x f x x lim , cos x 31. ( ) ( ) = → f x f x e lim , x x 0 32. ( ) ( ) = → f x f x x lim , ln x 1 33. ( ) ( ) = →− f x f x x lim , 1 x 1 34. ( ) ( ) = → f x f x x lim , 1 x 2 2 35. ( ) ( ) = ≥ < ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x x x x x lim , if 0 2 if 0 x 0 2 36. ( ) ( ) = − < − ≥ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x x x x x lim , 1 if 0 3 1 if 0 x 0 37. ( ) ( ) = ≤ + > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x x x x x lim , 3 if 1 1 if 1 x 1 38. ( ) ( ) = ≤ − > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x x x x x lim , if 2 2 1 if 2 x 2 2 39. ( ) ( ) = < = > ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ → f x f x x x x x x lim , if 0 1 if 0 3 if 0 x 0 40. ( ) ( ) = < − > ⎧ ⎨ ⎪ ⎩⎪⎪ → f x f x x x lim , 1 if 0 1 if 0 x 0 41. ( ) ( ) = ≤ > ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x x x x x lim , sin if 0 if 0 x 0 2 42. ( ) ( ) = > − ≤ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ → f x f x e x x x lim , if 0 1 if 0 x x 0 In Problems 43–48, use a graphing utility to investigate the indicated limit. Round answers to two decimal places. 43. − + − − + − → x x x x x x lim 1 2 2 x 1 3 2 4 3 44. + + + + + + →− x x x x x x lim 3 3 2 2 x 1 3 2 4 3 45. − + − + − → x x x x x lim 2 4 8 6 x 2 3 2 2 46. − + − + − → x x x x x lim 3 3 3 4 x 1 3 2 2 47. + + + + + →− x x x x x x lim 2 2 2 x 1 3 2 4 3 48. − + − − + − → x x x x x x lim 3 4 12 3 3 x 3 3 2 4 3
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