Finding Limits Graphically and Numerically

Lesson 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?

1 2

1 2

Average Velocity =

d dd

t t t

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

change in distance ( ) (2) feet

change in time 2 sec

s t s

t

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

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

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

Can be observed on a graph.

Observing a Limit

ViewDemoViewDemo

Observing a Limit

Can be observed on a graph.

Observing a Limit

Can be observed in a table

The limit is observed to be 64

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( )

2y x

x

Non Existent Limits

Note that f(x) does NOT get closer to a particular value• it grows without boundgrows without bound

There is NO LIMIT

Try command oncalculator

Non Existent Limits

f(x) grows without bound

View Demo3View

Demo3

Non Existent Limits

View Demo 4View

Demo 4

Formal Definition of a Limit

The

For any ε (as close asyou want to get to L)

There exists a (we can get as close as necessary to c )

lim ( )x cf x L

L •

c

View Geogebra View Geogebra demodemo

View Geogebra View Geogebra demodemo

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 …

( )f x L when x c

Specified Epsilon, Required Delta

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

4lim(2 7) 1x

x

2

Assignment

Lesson 2.2 Page 76 Exercises: 1 – 35 odd