950 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function In Words The limit of the quotient of two functions equals the quotient of their limits, provided that the limit of the denominator is not zero. Finding the Limit of a Power or a Root Find: (a) ( ) − → x lim 3 5 x 1 4 (b) + → x lim 5 8 x 0 2 (c) ( ) − + →− x x lim 5 3 x 1 3 4 3 Solution EXAMPLE 8 (a) ( ) ( ) ( ) − = ⎡ − ⎣⎢ ⎤ ⎦⎥ = − = → → x x lim 3 5 lim 3 5 2 16 x x 1 4 1 4 4 (b) ( ) + = + = = → → x x lim 5 8 lim 5 8 8 2 2 x x 0 2 0 2 (c) ( ) ( ) ( ) ( ) − + = − + = − + ⎡ ⎣⎢ ⎤ ⎦⎥ = − = = →− →− →− x x x x x x lim 5 3 lim 5 3 lim 5 3 1 1 1 x x x 1 3 4 3 1 3 4 3 1 3 4 3 4 3 3 Now Work PROBLEM 27 4 Find the Limit of a Quotient THEOREM Limit of a Quotient ( ) ( ) ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = → → → f x g x f x g x lim lim lim x c x c x c provided that ( ) ( ) → → f x g x lim and lim x c x c both exist, and ( ) ≠ → g x lim 0. x c Finding the Limit of a Quotient Find: − + + → x x x lim 5 2 3 4 x 1 3 Solution EXAMPLE 9 The limit involves the quotient of two polynomial functions: ( ) = − + f x x x 5 2 3 and ( ) = + g x x3 4. First, find the limit of the denominator ( ) g x . ( ) ( ) = + = → → g x x lim lim 3 4 7 x x 1 1 Since the limit of the denominator is not zero, use the Limit of a Quotient. ( ) ( ) − + + = − + + = → → → x x x x x x lim 5 2 3 4 lim 5 2 lim 3 4 6 7 x x x 1 3 1 3 1 Now Work PROBLEM 25 When the limit of the denominator is zero, the Limit of a Quotient cannot be used. In such cases, other strategies are needed. Let’s look at two examples. Finding the Limit of a Quotient Find: (a) − − − → x x x lim 6 9 x 3 2 2 (b) − − → x x lim 3 3 x 3 EXAMPLE 10 Solution (a) The limit of the denominator equals zero, so the Limit of a Quotient cannot be used. Instead, notice that the expression can be factored as ( )( ) ( )( ) − − − = − + − + x x x x x x x 6 9 3 2 3 3 2 2
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