section 1.2 - finding limits graphically and numerically
DESCRIPTION
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)…. - PowerPoint PPT PresentationTRANSCRIPT
Section 1.2 - Finding Limits Graphically and Numerically
LimitInformal 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.
limx cf x L
The limit of f(x)…
as x approaches c…
is L.
Notation:
c
L
f(x)
x
Calculating Limits
Our book focuses on three ways:
1. Numerical Approach – Construct a table of values
2. Graphical Approach – Draw a graph
3. Analytic Approach – Use Algebra or calculus
This Lesson
Next Lesson
Example 1Use the graph and complete the table to find the limit (if it exists).
3
2limxx
3
2limxx
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) 6.859 7.88 7.988 8 9.2618.128.012
If the function is continuous at the value of x, the limit is easy to calculate.
8
Example 2Use the graph and complete the table to find the limit (if it exists).
2 111
lim xxx
2 111
lim xxx
x -1.1 -1.01 -1.001 -1 -.999 -.99 -.9
f(x) -2.1 -2.01 -2.001 DNE -1.9-1.99-1.999
If the function is not continuous at the value of x, a graph and table can be very useful.
2
Can’t divide by 0
-6
Example 3Use the graph and complete the table to find the limit (if it exists).
4
7 if 4
lim if 8 if 4
1 if 4x
x x
f x f x x
x x
4
limx
f x
x -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9
f(x) 2.9 2.99 2.999 8 2.92.992.999
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.
3The limit does not change if the
value at -4 changes.
-6
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.
4
lim Does Not Existx
f x
Three Limits that Fail to Exist
f(x) increases or decreases without bound as x approaches c.
0
lim Does Not Existxf x
Three Limits that Fail to Exist
f(x) oscillates between two fixed values as x approaches c.
1
0limsin Does Not Existxx
x 0f(x) -1 1 -1 DNE 1 -1 1
2 2
3 25 2
52
32
Close
CloserClosest
A Limit that DOES ExistIf the domain is restricted (not infinite), the
limit of f(x) exists as x approaches an endpoint of the domain.
5
lim 5x
f x
Example 1Given the function t defined by the graph, find the limits at right.
4
3
0
6
2
5
1. lim
2. lim
3. lim
4. lim
5. lim
6. lim
x
x
x
x
x
x
t x
t x
t x
t x
t x
t x
2
3
DNE
3
DNE
2
t x
Example 2
Sketch a graph of the function with the following characteristics:
1. does not exist, Domain: [-2,3),
and Range: (1,5)
2. does not exist,
Domain: (-∞,-4)U(-4,∞), and
Range: (-∞,∞)
limx 0
f x
limx 0
f x
ClassworkSketch a graph and complete the table to find the limit (if it exists).
1
0lim 1
x
xx
1
0lim 1
x
xx
x -0.1 -0.01 -0.001 0 0.001 0.01 0.1
f(x) 2.8680 2.732 2.7196 DNE 2.59372.70482.7169
This a very important value that we will investigate more in Chapter 5. It deals with
natural logs.
e
Why is there a lot of “noise” over here?