SECTION 14.2 Algebraic Techniques for Finding Limits 951 Finding Limits Using Algebraic Properties Find: − + − − + − → x x x x x x lim 2 4 8 2 2 x 2 3 2 4 3 Solution EXAMPLE 11 The limit of the denominator is zero, so the Limit of a Quotient cannot be used. Factor the expression, and simplify. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − + − − + − = − + − − + − = + − + − x x x x x x x x x x x x x x x x 2 4 8 2 2 2 4 2 2 1 2 4 2 1 2 3 2 4 3 2 3 2 3 ↑ Factor by grouping Then x x x x x x x x x x lim 2 4 8 2 2 lim 4 2 1 2 8 9 x x 2 3 2 4 3 2 2 3 ( ) ( ) ( ) ( ) − + − − + − = + − + − = → → Finding the Limit of an Average Rate of Change Find the limit as x approaches 2 of the average rate of change of the function ( ) = + f x x x3 2 from 2 to x. Solution EXAMPLE 12 The average rate of change of f from 2 to x is ( ) ( ) ( ) ( )( ) − − = + − − = + − − f x f x x x x x x x 2 2 3 10 2 5 2 2 2 The limit of the average rate of change is f x f x x x x x x x lim 2 2 lim 3 10 2 lim 5 2 2 7 x x x 2 2 2 2 ( ) ( ) ( ) ( )( ) − − = + − − = + − − = → → → TIP In both of these examples we are able to “remove the division by zero” through an algebraic process. This is not always possible. j When computing a limit as x approaches 3, we are interested in the values of the function when x is close to 3 but unequal to 3. Since ≠ x 3, we can divide out the ( ) −x 3 ’s. Then, the Limit of a Quotient can be used. ( )( ) ( ) ( ) ( ) ( ) − − − = − + − + = + + = → → → → x x x x x x x x x lim 6 9 lim 3 2 3 3 lim 2 lim 3 5 6 x x x x 3 2 2 3 3 3 (b) Again, the limit of the denominator is zero.The strategy here is to rationalize the numerator. ( ) ( ) − − = − − ⋅ + + = − − + = + x x x x x x x x x x 3 3 3 3 3 3 3 3 3 1 3 ↑ ≠ x 3 Now since the limit of the denominator is not equal to zero, we can use the Limit of a Quotient. ( ) − − = + = + = + = = → → → → x x x x lim 3 3 lim 1 3 lim1 lim 3 1 3 3 1 2 3 3 6 x x x x 3 3 3 3 Compare the exact solution above with the approximate solution found in Example 4 of Section 14.1. Now Work PROBLEM 37 5 Find the Limit of an Average Rate of Change Now Work PROBLEM 43
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