1. Finding Limits Graphically and Numerically
Lesson 2.2

2. Average Velocity
Average velocity is the distance traveled divided by an elapsed time.
A boy rolls down a hill on a skateboard. At time = 4 seconds, the boy has rolled 6 meters from the top of the hill. At time = 7 seconds, the boy has rolled to a distance of 30 meters. What is his average velocity?

3. Distance Traveled by an Object
Given distance s(t) = 16t2
We seek the velocity
or the rate of change of distance
The average velocity between 2 and t
2 t

4. Average Velocity Use calculator
Graph with window 0 < x < 5, 0 < y < 100
Trace for x = 1, 3, 1.5, 1.9, 2.1, and then x = 2
What happened?
This is the average velocity function

5. Limit of the Function Try entering in the expression limit(y1(x),x,2)
The function did not exist at x = 2
but it approaches 64 as a limit
Expression
variable to get close
value to get close to

6. Limit of the Function Note: we can approach a limit from
left … right …both sides
Function may or may not exist at that point
At a
right hand limit, no left
function not defined
At b
left handed limit, no right
function defined a b

7. Observing a Limit
Can be observed on a graph.
View Demo

8. Observing a Limit
Can be observed on a graph.

9. Observing a Limit Can be observed in a table
The limit is observed to be 64

10. Non Existent Limits
Limits may not exist at a specific point for a function
Set
Consider the function as it approaches x = 0
Try the tables with start at –0.03, dt = 0.01
What results do you note?

11. Non Existent Limits
Note that f(x) does NOT get closer to a particular value
it grows without bound
There is NO LIMIT
Try command on calculator

12. Non Existent Limits
f(x) grows without bound
View
Demo3

13. Non Existent Limits
View Demo 4

14. Formal Definition of a Limit
The
For any ε (as close as you want to get to L)
There exists a  (we can get as close as necessary to c ) •
View Geogebra demo

15. Formal Definition of a Limit
For any  (as close as you want to get to L)
There exists a  (we can get as close as necessary to c Such that …

16. Specified Epsilon, Required Delta

17. Finding the Required  Consider showing
|f(x) – L| = |2x – 7 – 1| = |2x – 8| < 
We seek a  such that when |x – 4| < 
|2x – 8|<  for any  we choose
It can be seen that the  we need is

18. Assignment
Lesson 2.2
Page 76
Exercises: 1 – 35 odd