1.2 Finding Limits Graphically and Numerically
Objective: You will be able to:
• define a limit
• find a limit graphically and numerically
• determine the 3 ways a limit fails to exist
Aug 20­8:58 AM
You will be able to: define a limit find a limit graphically and numerically
In this class, we will frequently GNAW on functions.
This means functions can be represented in 4 ways:
Graphically
Numerically (table of values)
Algebraically (equation)
Words (verbally)
Aug 31­5:07 PM
1
You will be able to: define a limit find a limit graphically and numerically
What is a limit?
If f(x) becomes arbitrarily close to a single number L
as x approaches c from either side, the limit of f(x), as x approaches c, is L.
lim f(x) = L
x­>c
Aug 20­9:00 AM
You will be able to: define a limit find a limit graphically and numerically
Find: lim f(x)
x­>1
Graphically
(standard window)
Numerically
x
.75
.9
.999 1
1.001 1.01 1.1
1.25
f(x)
Words
As x approaches _____ from the left and the right, the limit of the function is: _____
Aug 20­9:04 AM
2
You will be able to: define a limit find a limit graphically and numerically
Aug 20­9:05 AM
You will be able to: define a limit find a limit graphically and numerically
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. Find: lim f(x)
x­>0
Aug 20­10:07 AM
3
You will be able to: define a limit find a limit graphically and numerically
A real life example of a limit....
Who would like to leave class early?
Does a function always equal it's limit value?
This idea should help us understand #12
lim f(x) =
x -> 1
f(1) =
Aug 20­9:04 AM
http://www.calculus­help.com/funstuff/tutorials/limits/limit01.html
Aug 30­11:12 AM
4
You will be able to: determine the 3 ways a limit fails to exist
Using your calculator, explore the limits of the following functions Graphically and / or Numerically
1.
3.
2.
Aug 31­5:10 PM
You will be able to: determine the 3 ways a limit fails to exist
Limits That Fail to Exist
1. Behavior That Differs From the Right and Left
Aug 31­5:10 PM
5
Objective: You will learn the 3 ways a limit fails to exist
Limits That Fail to Exist
2. Unbounded Behavior
Aug 31­5:10 PM
Objective: You will learn the 3 ways a limit fails to exist
Limits That Fail to Exist
3. Oscillating Behavior
Aug 31­5:10 PM
6
Objective: You will learn the 3 ways a limit fails to exist
http://www.calculus­help.com/funstuff/tutorials/limits/limit02.html
Aug 31­5:10 PM
Estimate the Limit Graphically
Aug 31­5:10 PM
7
Estimate a Limit Graphically
Aug 31­5:23 PM
You will be able to: define a limit find a limit graphically and numerically,
and learn the 3 ways a limit fails to exist
Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.
a)
b)
c)
d)
Aug 31­5:23 PM
8
You will be able to: define a limit find a limit graphically and numerically,
and learn the 3 ways a limit fails to exist
Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.
e) f) g)
h)
Aug 31­5:25 PM
You will be able to: define a limit find a limit graphically and numerically,
and learn the 3 ways a limit fails to exist
Sketch a graph of a function f that satisfies the given values. Aug 31­5:26 PM
9
You will be able to: define a limit find a limit graphically and numerically
In your own words...
1. Define a limit
2. What are the 3 cases when a limit fails to exist?
Aug 20­10:11 AM
Aug 20­10:23 AM
10