SECTION 14.2 Algebraic Techniques for Finding Limits 947 See Figure 6. For any number c , as x gets closer to c , the corresponding value of f is x , which is just as close to c . That is, = → x c lim . x c Figure 6 = → x c lim x c f(x) 5 x c c x y (c, c) In Words The limit of the sum of two functions equals the sum of their limits. Finding the Limit of a Sum Find: x lim 4 x 3 ( ) + →− Solution EXAMPLE 2 The limit involves the sum of two functions: ( ) = f x x and ( ) = g x 4. From the basic limits (1) and (2), ( ) ( ) = = − = = →− →− →− →− f x x g x lim lim 3 and lim lim 4 4 x x x x 3 3 3 3 Then using the Limit of a Sum theorem, it follows that ( ) + = + = − + = →− →− →− x x lim 4 lim lim4 3 4 1 x x x 3 3 3 In Words The limit of the difference of two functions equals the difference of their limits. Finding the Limit of a Difference Find: ( ) − → x lim 6 x 4 EXAMPLE 3 Using the Two Basic Limits (a) = → lim5 5 x 3 (b) = → x lim 3 x 3 (c) ( ) − = − → lim 8 8 x 0 (d) = − →− x lim 1 2 x 1 2 EXAMPLE 1 Now Work PROBLEM 9 Formulas (1) and (2), when used with the properties that follow, enable us to find limits of more complicated functions. 1 Find the Limit of a Sum, a Difference, and a Product In the following properties, assume that f and g are two functions for which both ( ) → f x lim x c and ( ) → g x lim x c exist. THEOREM Limit of a Sum [ ] ( ) ( ) ( ) ( ) + = + → → → f x g x f x g x lim lim lim x c x c x c THEOREM Limit of a Difference [ ] ( ) ( ) ( ) ( ) − = − → → → f x g x f x g x lim lim lim x c x c x c (continued)
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