952 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function SUMMARY To find exact values for ( ) → f x lim , x c try using basic limits and algebraic properties of limits. • If f is a polynomial function, ( ) ( ) = → f x f c lim x c . • If f is a polynomial raised to a power or is the root of a polynomial, use the Limit of a Power or a Root with the Limit of a Polynomial. • If f is a quotient and the limit of the denominator is not zero, use the Limit of a Quotient. • If f is a quotient and the limit of the denominator is zero, try other techniques, such as factoring or rationalizing. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 14.2 Assess Your Understanding 1. Find the average rate of change of ( ) = f x x2 from 2 to x . (pp. 93–94) 2. Rationalize the numerator of the quotient: − − x x 5 5 (pp. A89–A90) Concepts and Vocabulary 3. The limit of the product of two functions equals the of their limits. 4. A lim x c = → . 5. Multiple Choice x lim x c = → . (a) x (b) c (c) cx (d) x c 6. True or False The limit of a polynomial function as x approaches 5 equals the value of the polynomial at 5. 7. True or False The limit of a rational function at 5 equals the value of the rational function at 5. 8. True or False The limit of a quotient equals the quotient of the limits. Skill Building In Problems 9–42, find each limit algebraically. 9. → lim5 x 1 10. ( ) − → lim 3 x 1 11. → x lim x 4 12. →− x lim x 3 13. ( ) →− x lim 5 x 2 14. ( ) − → x lim 3 x 4 15. ( ) + → x lim 3 2 x 2 16. ( ) − → x lim 2 5 x 3 17. ( ) − →− x x lim 3 5 x 1 2 18. ( ) − → x lim 8 4 x 2 2 19. ( ) − + − → x x x lim 5 3 6 9 x 1 4 2 20. ( ) − + + − →− x x x x lim 8 7 8 4 x 1 5 3 2 21. ( ) + → x lim 1 x 1 2 3 22. ( ) − → x lim 3 4 x 2 2 23. + → x lim 5 4 x 1 24. − → x lim 1 2 x 0 25. − + → x x lim 4 4 x 0 2 2 26. + + → x x x lim 3 4 x 2 2 27. ( ) − → x lim 3 2 x 2 5 2 28. ( ) + →− x lim 2 1 x 1 5 3 29. − − → x x x lim 4 2 x 2 2 2 30. + − →− x x x lim 1 x 1 2 2 31. − − − →− x x x lim 12 9 x 3 2 2 32. + − + − →− x x x x lim 6 2 3 x 3 2 2 33. − − → x x lim 1 1 x 1 3 34. − − → x x lim 1 1 x 1 4 35. ( ) + − →− x x lim 1 1 x 1 2 2 36. − − → x x lim 8 4 x 2 3 2 37. − + − + − → x x x x x lim 2 4 8 6 x 2 3 2 2 38. − + − + − → x x x x x lim 3 3 3 4 x 1 3 2 2 39. + + + + + →− x x x x x x lim 2 2 2 x 1 3 2 4 3 40. x x x x x x lim 3 4 12 3 3 x 3 3 2 4 3 − + − − + − → 41. − − → x x lim 2 2 x 2 42. − − → x x lim 5 5 x 5 In Problems 43–52, find the limit as x approaches c of the average rate of change of each function from c to x. 43. ( ) = = − c f x x 2; 5 3 44. ( ) = − = − c f x x 2; 4 3 45. ( ) = = c f x x 3; 2 46. ( ) = = c f x x 3; 3 47. ( ) = − = + c f x x x 1; 2 2 48. ( ) = − = − c f x x x 1; 2 3 2 49. ( ) = = − + c f x x x 0; 3 2 4 3 2 50. ( ) = = − + c f x x x 0; 4 5 8 3 51. ( ) = = c f x x 1; 1 52. ( ) = = c f x x 1; 1 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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