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

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

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

Looks like y=1

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

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

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

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

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.

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

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

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