SECTION 14.3 One-sided Limits; Continuity 953 Applications and Extensions In Problems 53–56, use the properties of limits and the facts that = − = = = → → → → x x x x x x lim sin 1 lim cos 1 0 lim sin 0 lim cos 1 x x x x 0 0 0 0 where x is in radians, to find each limit. 53. → x x lim tan x 0 54. ( ) → x x lim sin 2 x 0 [Hint : Use a Double-angle Formula.] 55. + − → x x x lim 3sin cos 1 4 x 0 56. ( ) + − → x x x x lim sin sin cos 1 x 0 2 2 Retain Your Knowledge Problems 57–60 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for a final exam, or subsequent courses such as calculus. 57. Graph the function ( ) = + + f x x x 1 3 2 . 58. Find the inverse of the function ( ) = + + g x x x 2 3 1 . 59. Give the exact value of − sin 3 2 1 . 60. Use the Binomial Theorem to expand ( ) +x 2 4 . ‘Are You Prepared?’ Answers 1. x x x 2 2 2 ( )( ) − + − 2. x 1 5 + 14.3 One-sided Limits; Continuity Now Work the ‘Are You Prepared?’ problems on page 958. • Library of Functions (Section 2.4, pp. 100–104) • Piecewise-defined Functions (Section 2.4, pp. 105–107) • Polynomial Functions (Section 4.1, pp. 190–20 1) • Properties of Rational Functions (Section 4.5, pp. 236–244) • The Graph of a Rational Function (Section 4.6, pp. 247–254) • Properties of the Exponential Function (Section 5.3, pp. 294–30 1) • Properties of the Logarithmic Function (Section 5.4, p. 316) • Properties of the Trigonometric Functions (Section 6.4, pp. 428 and 430, and Section 6.5, pp. 443–445 and 447) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find the One-sided Limits of a Function (p. 953) 2 Determine Whether a Function Is Continuous at a Number (p. 955) 1 Find the One-sided Limits of a Function Earlier we described f x N lim x c ( ) = → by saying that as x gets closer to the number c but remains unequal to c , the corresponding values of f get closer to the number N . Whether we use a numerical argument or the graph of the function f, the variable x can get closer to c in only two ways: by approaching c from the left, using numbers less than c , or by approaching c from the right, using numbers greater than c . If we approach c from only one side, we have a one-sided limit. The notation f x L lim x c ( ) = → −
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