12.2 Techniques For Calculating Limits

1. 12.2 Techniques For Calculating Limits • Rules for Limits • Constant rule If k is a constant real number, • Limit of x rule For the following rules, we assume that and both exist • Sum and difference rules

2. 12.2 Techniques For Calculating Limits • Rules for Limits • Product Rule • Quotient Rule • provided

3. 12.2 Finding a Limit of a Linear Function Example Find Solution Rules 1 and 4 Rules 1 and 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. 12.2 Finding a Limit of a Polynomial Function with One Term For any polynomial function in the form

6. 12.2 Finding a Limit of a Polynomial Function Example Find . Solution Rule 3

7. 12.2 Techniques For Calculating Limits • Rules for Limits (Continued) • For the following rules, we assume that and • both exist. • Polynomial rule If p(x) defines a polynomial function, then

8. 12.2 Techniques For Calculating Limits • Rules for Limits (Continued) • 7. Rational function rule If f(x) defines a rational • function with then • Equal functions rule If f(x) = g(x)for all , then

9. 12.2 Techniques For Calculating Limits Rules for Limits (Continued) 9. Power rule For any real number k, provided this limit exists.

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. 12.2 Finding a Limit of a Rational Function Example Find Solution Rule 7 cannot be applied directly since the denominator is 0. First factor the numerator and denominator

12. 12.2 Finding a Limit of a Rational Function Solution Now apply Rule 8 with and so that f(x) = g(x) for all .

13. 12.2 Finding a Limit of a Rational Function Solution Rule 8 Rule 6