1. Copyright © 2011 Pearson Education, Inc. Slide 12.2-1 12.2 Techniques For Calculating Limits Rules for Limits 1.Constant rule If k is a constant real number, 2.Limit of x rule For the following rules, we assume that and both exist 3.Sum and difference rules

2. Copyright © 2011 Pearson Education, Inc. Slide 12.2-2 12.2 Techniques For Calculating Limits Rules for Limits 4.Product Rule 5.Quotient Rule provided

3. Copyright © 2011 Pearson Education, Inc. Slide 12.2-3 12.2 Finding a Limit of a Linear Function Example Find Solution Rules 1 and 4 Rules 1 and 2

4. Copyright © 2011 Pearson Education, Inc. Slide 12.2-4 12.2 Finding a Limit of a Polynomial Function with One Term Example Find Solution Rule 4 Rule 1 Rule 4 Rule 2

5. Copyright © 2011 Pearson Education, Inc. Slide 12.2-5 12.2 Finding a Limit of a Polynomial Function with One Term For any polynomial function in the form

6. Copyright © 2011 Pearson Education, Inc. Slide 12.2-6 12.2 Finding a Limit of a Polynomial Function Example Find. Solution Rule 3

7. Copyright © 2011 Pearson Education, Inc. Slide 12.2-7 12.2 Techniques For Calculating Limits Rules for Limits (Continued) For the following rules, we assume that and both exist. 6.Polynomial rule If p(x) defines a polynomial function, then

8. Copyright © 2011 Pearson Education, Inc. Slide 12.2-8 12.2 Techniques For Calculating Limits Rules for Limits (Continued) 7.Rational function rule If f(x) defines a rational function with then 8.Equal functions rule If f(x) = g(x) for all, then

9. Copyright © 2011 Pearson Education, Inc. Slide 12.2-9 12.2 Techniques For Calculating Limits Rules for Limits (Continued) 9.Power rule For any real number k, provided this limit exists.

10. Copyright © 2011 Pearson Education, Inc. Slide 12.2-10 12.2 Techniques For Calculating Limits Rules for Limits (Continued) 10.Exponent rule For any real number b > 0, 11.Logarithm rule For any real number b > 0 with, provided that

11. Copyright © 2011 Pearson Education, Inc. Slide 12.2-11 12.2 Finding a Limit of a Rational Function Example Find SolutionRule 7 cannot be applied directly since the denominator is 0. First factor the numerator and denominator

12. Copyright © 2011 Pearson Education, Inc. Slide 12.2-12 12.2 Finding a Limit of a Rational Function SolutionNow apply Rule 8 with and so that f(x) = g(x) for all.

13. Copyright © 2011 Pearson Education, Inc. Slide 12.2-13 12.2 Finding a Limit of a Rational Function Solution Rule 8 Rule 6