1. Bell-ringer 9/21/09 Graph the function:
Use the following window:
XMIN: YMIN: -2
XMAX: 2 YMAX: 2
Xscale: /2 Yscale: 1

2. Graphical Approach to Limits
Grand Teton National Park, Wyoming
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2007

3. Consider:
What happens as x approaches zero?
Graphically: Y=
WINDOW
GRAPH

4. Looks like y=1

5. Numerically:
TblSet
TABLE
You can scroll down to see more values.

6. It appears that the limit of as x approaches zero is 1
TABLE
You can scroll down to see more values.

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

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

9. Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
(See your book 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.

10. does not exist because the left and right hand limits do not match!2 1 1 2 3 4
At x=1:
left hand limit
right hand limit
value of the function

11. because the left and right hand limits match.2 1 1 2 3 4
At x=2:
left hand limit
right hand limit
value of the function

12. because the left and right hand limits match.2 1 1 2 3 4
At x=3:
left hand limit
right hand limit
value of the function