1. 2.1 day 1: Step Functions “Miraculous Staircase” Loretto Chapel, Santa Fe, NM Two 360 o turns without support!

2. Limit notation: “The limit of f of x as x approaches c is L.” So:

3. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

4. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 58 for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

5. 1234 1 2 At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match!

6. At x=2:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

7. At x=3:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

8. “Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function. The TI-83 contains the command, but it is important that you understand the function rather than just entering it in your calculator.

9. Greatest Integer Function:

13. The greatest integer function is also called the floor function. The notation for the floor function is: Some books use or.

14. The calculator “connects the dots” which covers up the discontinuities. The TI-83 command for the floor function is int (x). Graph the floor function for and. Y= CATALOG I int( int (x)

15. Go toY= Highlight the slant line to the left. ENTER GRAPH The open and closed circles do not show, but we can see the discontinuities. The TI-83 command for the floor function is int (x). Graph the floor function for and. Pressuntil the line appears dotted

16. Least Integer Function:

20. The least integer function is also called the ceiling function. The notation for the ceiling function is: Least Integer Function: The TI-89 command for the ceiling function is ceiling (x). Don’t worry, there are not wall functions, front door functions, fireplace functions!