1. What you’ll learn about
Average and Instantaneous Speed
Definition of Limit
Properties of Limits
One-Sided and Two-Sided Limits
Sandwich Theorem
The Limit concept is a calculus concept. Limits can be used to describe continuity, the Derivative and the Integral: the ideas giving the foundation of calculus.
In Ch2 we will define and calculate Limits, by substitution, numerically, analytically, graphical investigation – or a combination of these.

2. Average Rate of Change vs Instantaneous Rate of Change
Let’s begin our discussion of the difference between the Average rate of change (ROC) and the Instantaneous ROC by first reviewing the slope of the graph of a linear function.
The slope of a line is constant! That is, Δx and Δy don’t change as you move on the graph.
Let’s do a real world example.

3. 3. Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:
But, how do we set this up using points on a graph? If
4. In this application the slope is constant, but, how do you find the slope of a curve, where the slope varies?

4. 4. If you look at your speedometer during this trip at a particular time, it might read 65 mph. This is your instantaneous speed.
A real world example: A rock falls from a high cliff. Find the instantaneous speed after 2 seconds.
The position of the rock at any time t is given by the function:
At t = 2 sec.:
average speed:
32 is the average speed, but what is the instantaneous speed at 2 seconds?

5. Why can you NOT use the average speed method to find the instantaneous speed?
for some very small change in t
where h = some very small change in t, or
We can use a graphing calculator to evaluate this expression for smaller and smaller values of h. You can see this in the book on page 60.

6. We can see that the velocity approaches 64 ft/sec as h becomes very small.1 80
0.1
65.6
.01
64.16
.001
64.016
.0001
.00001
We say that the velocity has a limiting value of 64 as h approaches zero.
(Note that h never actually becomes zero.)

7. Now, let’s find the limit Algebraically (Analytically)
The limit as h approaches zero:
Since the 16 is unchanged as h approaches zero, we can factor 16 out.

8. Okay now, Let’s find the limit of:
First, we cannot use substitution; can you see why?
So, we do this graphicaally
What happens as x approaches zero? Y=
WINDOW
GRAPH

9. Looks like y=1

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

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

12. Okay what is the the formal and informal definitions of a Limit
Okay what is the the formal and informal definitions of a Limit. For the formal definition see page 60. The informal definition is as follows:
Limit notation:
“The limit of f of x as x approaches c is L.”
The notation means that the values f (x) of the
function f approach or equal L as the values
of x approach (but do not equal) c.
So:

13. Example Limits

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

15. Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. See book for details. Let’s do some examples:
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.

16. One-Sided and Two-Sided Limits

17. 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

18. 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

19. 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

20. The Sandwich Theorem: Show that: The maximum value of sine is 1, so
The minimum value of sine is -1, so
So:

21. By the sandwich theorem:
WINDOW

22. p