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?
Average Velocity =
d d1  d 2

t
t1  t2
Distance Traveled by an Object
 Given distance s(t) = 16t2
 We seek the velocity
• or the rate of change of distance
2 t
 The average velocity between 2 and t
change in distance s(t )  s(2) feet


change in time
t 2
sec
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)
Expression
variable to
get close
value to get
close to
 The function did not exist at x = 2
• but it approaches 64 as a limit
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
Observing a Limit
 Can be observed on a graph.
View
Demo
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
1
y1( x) 
Set
2x
Consider the function as it approaches
x=0
Try the tables with start at –0.03, dt = 0.01
What results do you note?
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
Non Existent Limits
 f(x) grows without bound
View
Demo3
Non Existent Limits
View
Demo 4
Formal Definition of a Limit
 The
lim f ( x)  L
x c
L 
•
 For any ε (as close as
c 
you want to get to L)
 There exists a  (we can get as close as
necessary to c )
View Geogebra
demo
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
lim(2 x  7)  1
x4
 |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

2
Assignment
 Lesson 2.2
 Page 76
 Exercises: 1 – 35 odd