Evaluating Limits Analytically

1. Evaluating Limits Analytically Section 12.2(day1)

2. After this lesson, you will be able to: • evaluate a limit using the properties of limits • develop and use a strategy for finding limits

3. Limits Analytically In the previous lesson, you learned how to find limits numerically and graphically. In this lesson you will be shown how to find them analytically…using algebra or calculus.

4. Some Basic Limits Let b and c be real numbers and let n be a positive integer. Examples: Think of it graphically… Let Let Let (y scale was adjusted to fit) As x approaches 5, f(x) approaches 125 As x approaches 3, f(x) approaches 4 As x approaches 2, f(x) approaches 2

5. Direct Substitution • Some limits can be evaluated by direct substituting for x. • Direct substitution works on continuous functions. • Continuous functions do not have any holes, breaks or gaps. Note: Direct substitution is valid for all polynomial functions and rational functions whose denominators are not zero.

6. Limit of a Polynomial Function 5 Example: Since a polynomial function is a continuous function, then we know the limit from the right and left of any number will be the same. Thus, we may use direct substitution.

7. Limit of a Rational Function -8 Example: Since this rational function has a nonzero denominator when x = 3, we may use direct substitution.

8. Limit of a Rational Function 4 Example: Since this rational function has a zero denominator when x = -2, we may NOT use direct substitution. Another method must be used.

9. Finding Limits • Try Direct Substitution • If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to divide out common factors, so that direct substitution works. • Use a graph or table to reinforce your result.

10. Example- Factoring Factor Example: Direct substitution at this point will give you 0 in the denominator. Using a bit of algebra, we can try to find the limit. Now direct substitution will work Graph on your calculator and use the table to check your result

11. Example- Factoring Example: Factor Direct substitution at this point will give you 0 in the denominator. Using a bit of algebra, we can try to find the limit. Now direct substitution will work Graph on your calculator and use the table to check your result

12. Example- Factoring Example: Direct substitution results in 0 in the denominator. Try factoring. Now direct substitution will work Use your calculator to reinforce your result

13. Example Sum of cubes Not factorable Example: Direct substitution results in 0 in the denominator. Try factoring. None of the factors can be divided out, so direct substitution still won’t work. The limit DNE. Verify the result on your calculator. The limits from the right and left do not equal each other, thus the limit DNE. Observe how the right limit goes to off to positive infinity and the left limit goes to negative infinity.

14. HOMEWORK: 12-2(DAY1) Worksheet