948 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function Solution The limit involves the difference of two functions: ( ) = f x 6 and ( ) = g x x. Using the two basic limits, ( ) ( ) = = = = → → → → f x g x x lim lim 6 6 and lim lim 4 x x x x 4 4 4 4 Then using the Limit of a Difference theorem, it follows that ( ) − = − = − = → → → x x lim 6 lim6 lim 6 4 2 x x x 4 4 4 In Words The limit of the product of two functions equals the product of their limits. THEOREM Limit of a Product [ ] ( ) ( ) ( ) ( ) ⋅ = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ → → → f x g x f x g x lim lim lim x c x c x c Finding the Limit of a Product Find: ( ) − →− x lim 4 x 5 Solution EXAMPLE 4 The limit involves the product of two functions: ( ) = − f x 4 and ( ) = g x x. ( ) ( ) ( ) = − = − = = − →− →− →− →− f x g x x lim lim 4 4 and lim lim 5 x x x x 5 5 5 5 Now using the Limit of a Product theorem, ( ) ( ) ( )( ) − = ⎡ − ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ = − − = →− →− →− x x lim 4 lim 4 lim 4 5 20 x x x 5 5 5 Finding Limits Using Algebraic Properties Find: (a) ( ) − →− x lim 3 5 x 2 (b) ( ) → x lim 5 x 2 2 Solution EXAMPLE 5 (a) ( ) ( ) ( ) − = − = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ − = ⋅ − − = − − = − →− →− →− →− →− →− x x x lim 3 5 lim 3 lim 5 lim 3 lim lim 5 3 2 5 6 5 11 x x x x x x 2 2 2 2 2 2 (b) ( ) ( ) = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ = ⋅ = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ = ⋅ ⋅ = → → → → → → x x x x x x lim 5 lim5 lim 5 lim 5 lim lim 5 2 2 20 x x x x x x 2 2 2 2 2 2 2 2 Now Work PROBLEM 15 Notice in the solution to part (b) that ( ) = ⋅ → x lim 5 5 2 . x 2 2 2 THEOREM Limit of a Monomial If ≥ n 1 is an integer and a is a constant, then for any real number c , ( ) = → ax ac lim x c n n Proof n n n factors factors factors ( ) ( ) = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ = ⋅ ⋅ ⋅ ⋅ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ ⎡ ⎣⎢ ⎤ ⎦⎥ = ⋅ ⋅ ⋅ ⋅ ⋅ = → → → → → → → → ax a x a x x x x a x x x x a c c c c ac lim lim lim lim . . . lim lim lim . . . lim . . . x c n x c x c n x c x c x c x c x c n Limit of a Product x c lim x c = → ■
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