Copyright 2008 Pearson Education Inc Publishing as Pearson
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1
1. 2 Algebraic Limits and Continuity OBJECTIVES Ø Develop and use the Limit Principles to calculate limits. Ø Determine whether a function is continuous at a point. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1. 2 Algebraic Limits and Continuity LIMIT PRINCIPLES: If and then we have the following: L. 1 The limit of a constant is the constant. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3
1. 2 Algebraic Limits and Continuity LIMIT PRINCIPLES (continued): L. 2 The limit of a power is the power of the limit, and the limit of a root is the root of the limit. That is, for any positive integer n, and assuming that L ≥ 0 when n is even. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4
1. 2 Algebraic Limits and Continuity LIMIT PRINCIPLES (continued): L. 3 The limit of a sum or difference is the sum or difference of the limits. L. 4 The limit of a product is the product of the limits. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5
1. 2 Algebraic Limits and Continuity LIMIT PRINCIPLES (concluded): L. 5 The limit of a quotient is the quotient of the limits. L. 6 The limit of a constant times a function is the constant times the limit. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6
1. 2 Algebraic Limits and Continuity Example 1: Use the limit principles to find We know that By Limit Principle L 4, Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7
1. 2 Algebraic Limits and Continuity Example 1 (concluded): By Limit Principle L 6, By Limit Principle L 1, Thus, using Limit Principle L 3, we have Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8
1. 2 Algebraic Limits and Continuity THEOREM ON LIMITS OF RATIONAL FUNCTIONS For any rational function F, with a in the domain of F, Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9
1. 2 Algebraic Limits and Continuity Example 3: Find Theorem on Limits of Rational Functions and Limit Principle L 2 tell us that we can substitute to find the limit: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 10
1. 2 Algebraic Limits and Continuity Example 4: Find Note that the Theorem on Limits of Rational Functions does not immediately apply because – 3 is not in the domain of However, if we simplify first, the result can be evaluated at x = – 3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 11
1. 2 Algebraic Limits and Continuity Example 4 (concluded): Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 12
1. 2 Algebraic Limits and Continuity DEFINITION: A function f is continuous at x = a if: 1) 2) 3) exists, (The output at a exists. ) exists, (The limit as exists. ) (The limit is the same as the output. ) A function is continuous over an interval if it is continuous at each point in that interval. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 13
1. 2 Algebraic Limits and Continuity Example 7: Is the function f given by continuous at x = 3? Why or why not? 1) 2) By the Theorem on Limits of Rational Functions, 3) Since f is continuous at x = 3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 14
1. 2 Algebraic Limits and Continuity Example 8: Is the function g given by continuous at x = – 2? Why or why not? 1) 2) To find the limit, we look at left and right-side limits. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 15
1. 2 Algebraic Limits and Continuity Example 8 (concluded): 3) Since we see that the does not exist. Therefore, g is not continuous at x = – 2. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 16
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