946 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function Retain Your Knowledge Problems 49–52 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. 49. Let ( ) = − A 2, 3 and ( ) = − B 6, 11 be points in the plane. Find the distance between the points and the midpoint of the line segment connecting the points. 50. Find the center, foci, and vertices of the ellipse ( ) ( ) − + + = x y 2 9 1 13 1 2 2 51. Logan invests $4000 at an annual interest rate of 6%. How much money will she have after 10 years if interest is compounded continuously? 52. Assuming > r 0 and θ π ≤ < 0 2 , find the polar coordinates of the point whose rectangular coordinates are ( ) −2, 2 3 . ‘Are You Prepared?’ Answers 1. See Figure 2 on page 943. 2. ( ) = f 0 0 OBJECTIVES 1 Find the Limit of a Sum, a Difference, and a Product (p. 947) 2 Find the Limit of a Polynomial (p. 949) 3 Find the Limit of a Power or a Root (p. 949) 4 Find the Limit of a Quotient (p. 950) 5 Find the Limit of an Average Rate of Change (p. 951) 14.2 Algebraic Techniques for Finding Limits Now Work the ‘Are You Prepared?’ problems on page 952. • Rationalize a Numerator (Section A.10, pp. A89–A90) • Average Rate of Change (Section 2.3, pp. 93–94) PREPARING FOR THIS SECTION Before getting started, review the following: In Section 14.1, we investigated limits using tables and graphs, and stated that algebra can sometimes be used to find the exact value of a limit. We state, without proof, two basic limits and several properties of limits. We use graphs to illustrate these two theorems. See Figure 5. Since the graph of a constant function is a horizontal line, it follows that no matter how close x is to c , the corresponding value of f equals A . That is, = → A A lim . x c THEOREM The Limit of a Constant If ( ) = f x A, where A is a constant, then for any real number c , ( ) = = → → f x A A lim lim x c x c (1) THEOREM The Limit of the Identity Function If ( ) = f x x, then for any real number c , ( ) = = → → f x x c lim lim x c x c (2) In Words The limit of a constant is the constant. In Words The limit of x as x approaches c is c . Figure 5 = → A A lim x c x y f (x) 5 A c (0, A)
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