Transcript Finding Limits Graphically and Numerically

Section 1.2 - Finding Limits
Graphically and Numerically
Limit
Informal Definition: If f(x) becomes arbitrarily close
to a single REAL number L as x approaches c
from either side, the limit of f(x), as x appraches
c, is L.
f(x)
L
x
c
The limit of f(x)…
Notation:
l
i
m
f
x
L
x

c
is L.
as x approaches c…
Calculating Limits
Our book focuses on three ways:
1. Numerical Approach – Construct a table
This of values
Lesson
2. Graphical Approach – Draw a graph
3. Analytic Approach – Use Algebra or
calculus
Next
Lesson
Example 1
Use the graph and complete the table to find the limit (if it exists).
3
limx
x2
x
1.9
1.99
1.999
2
2.001
2.01
2.1
f(x)
6.859
7.88
7.988
8
8.012
8.12
9.261
If the function is continuous at the value of x,
the limit is easy to calculate.
limx 8
3
x
2
Example 2
Use the graph and complete the table to find the limit (if it exists).
x21
x1
lim
x
1
Can’t divide by 0
x
-1.1
-1.01
-1.001
-1
-.999
-.99
-.9
f(x)
-2.1
-2.01
-2.001
DNE
-1.999
-1.99
-1.9
If the function is not continuous at the value of
x, a graph and table can be very useful.
lim 2
x2
1
x
1
x

1
Example 3
Use the graph and complete the table to find the limit (if it exists).
7

xi
f
x


4


l
i
m
f
xi
ff
x
8 i
f
x


4
-6

x


4


1

xi
f
x


4

x
-4.1
-4.01
-4.001
-4
-3.999
-3.99
-3.9
f(x)
2.9
2.99
2.999
-6
8
2.999
2.99
2.9
If the function is not continuous at the value of
x, the important thing is what the output gets
closer to as x approaches the value.
The limit does not change if
the value at -4 changes.
lim
fx
3

x


4
Three Limits that Fail to Exist
f(x) approaches a different number from the right
side of c than it approaches from the left side.
l
i
m
f
x

D
o
e
s
N
o
t
E
x
i
s
t


x


4
Three Limits that Fail to Exist
f(x) increases or decreases without bound
as x approaches c.
l
i
m
f
x

D
o
e
s
N
o
t
E
x
i
s
t


x

0
Three Limits that Fail to Exist
f(x) oscillates between two fixed values as x
approaches c.
Closest
Closer
Close
x
f(x)
 2  32  52 0 52
-1 1 -1 DNE 1
2
3
2

-1
1
1
l
i
m
s
i
nD

o
e
s
N
o
t
E
x
i
s
t


x
x

0
A Limit that DOES Exist
If the domain is restricted (not infinite), the
limit of f(x) exists as x approaches an
endpoint of the domain.
lim f  x   5
x 5
Example 1
Given the function t defined by the graph, find the limits at right.
tx
1. lim t  x  2
x4
2. lim t  x  3
x 3
3. lim t  x  
D

x0
4. lim t  x   3
x6
5. lim t  x  
D

x2
6. lim t  x  2
x 5

Example 2
Sketch a graph of the function with the
following characteristics:
1. lim
f
x does not exist, Domain: [-2,3),
x
0
and Range: (1,5)
2. lim
f
x does not exist,
x
0
Domain: (-∞,-4)U(-4,∞), and
Range: (-∞,∞)
Classwork
Sketch a graph and complete the table to find the limit (if it exists).
lim
1x
1x
x

0
Why is there a lot of
“noise” over here?
x
-0.1
-0.01
-0.001
0
0.001
0.01
0.1
f(x)
2.8680
2.732
2.7196
DNE
2.7169
2.7048
2.5937
This a very important value that we will
investigate more in Chapter 5. It deals with
natural logs.
l
im
1

x



e
x

0
1
x