Warm-up 1.2 Is there a value of the function at x = 1 ? Can you tell what the value “should” be?

1. Warm-up 1.2 Is there a value of the function at x = 1 ? Can you tell what the value “should” be?

2. Lesson 1.2Finding Limits Graphically & Numerically

3. Why limits? • Not all functions have y-values (or seem to have more than 1 y-value) at an x-value • Limits allow us to determine what the value “should” be → approaches • From the warm-up The limit of f(x) as x approaches 1 is 3.

4. Limits input output For now, we will calculate the limits numerically (table) or graphically

5. Example 1 Find the limit of the following:

6. Example 2

7. The actual value of f(x) at c has no bearing on the limit!

8. Limits Failing to Exist • f(x) approaches different numbers from left and right sides • f(x) approaches infinity as it approaches • f(x) oscillates between two numbers as it approaches

9. Example 3

10. Example 4

11. Example 5

12. Summary • Limit must approach a single value from left and right side to exist. • Limit value is equal to the number the limit approaches.

13. Graph the function f(x) = 2x – 5 What if someone told us we just need the function values to be within a certain range of the limit? Then, what would be the range of x values to keep the function values within this range? Could we have guessed the relationship between the errors from the equation?

14. Epsilon-Delta Definition of a Limit

16. Problem Set 1.2