1. Techniques for Evaluating Limits
Section 12.2
Techniques for Evaluating Limits

2. Objective By following instructions students will be able to:
Find the limits of polynomial and rational functions direct substitution.
Use the dividing out technique to find limits of functions.
Use the rationalizing technique to find limits of functions.
Approximate limits of functions graphically and numerically.
Evaluate one sided limits of functions.
Evaluate the limits of difference quotients from calculus.

3. Limits of Polynomial and Rational Functions.

4. EXAMPLE 1: Find each of the following limits by direct substitution.
a) b)

5. EXAMPLE 2:
Find the following limit by dividing out.

6. EXAMPLE 3:
Find the following limit by dividing out.

7. EXAMPLE 4:
Find the following limit by rationalizing.

8. EXAMPLE 5:
Approximate the limit using a graphing calculator.

9. U-TRY #1
Find the limit, if it exists. 1) 2)

10. EXAMPLE 6:
Approximate the limit using a graphing calculator.

11. EXAMPLE 7:
Find the limit as x0 from the left and the limit x0 from the right for

12. EXAMPLE 8:
Find the limit f(x) as x approaches 1.

13. EXAMPLE 9:
An overnight delivery service charges $8 for the first pound and $2 for each additional pound. Let x represent the weight of a parcel and let f(x) represent the shipping cost. Show that the limit of f(x) as x2 does not exist.

14. EXAMPLE 10:
For the function , find the difference quotient.

15. U-TRY #2
Find 1) 2)

16. Objective
Did we…
Find the limits of polynomial and rational functions direct substitution?
Use the dividing out technique to find limits of functions?
Use the rationalizing technique to find limits of functions?
Approximate limits of functions graphically and numerically?
Evaluate one sided limits of functions?
Evaluate the limits of difference quotients from calculus?

17. Homework
Pg 824 #s 1-59 EOO