1. Section 1.2 - Finding Limits Graphically and Numerically

2. 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. c L
f(x) x
The limit of f(x)…
is L.
Notation:
as x approaches c…

3. Calculating Limits Our book focuses on three ways:
Numerical Approach – Construct a table of values
Graphical Approach – Draw a graph
Analytic Approach – Use Algebra or calculus
This Lesson
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4. Example 1
Use the graph and complete the table to find the limit (if it exists). 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.

5. Example 2
Use the graph and complete the table to find the limit (if it exists).
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.

6. The limit does not change if the value at -4 changes.
Example 3
Use the graph and complete the table to find the limit (if it exists). -6 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.

7. 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.

8. Three Limits that Fail to Exist
f(x) increases or decreases without bound as x approaches c.

9. Three Limits that Fail to Exist
f(x) oscillates between two fixed values as x approaches c.
Closest
Closer
Close x
f(x) -1 1
DNE

10. 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.

11. Example 1
Given the function t defined by the graph, find the limits at right.

12. Example 2
Sketch a graph of the function with the following characteristics:
does not exist, Domain: [-2,3), and Range: (1,5)
does not exist, Domain: (-∞,-4)U(-4,∞), and Range: (-∞,∞)

13. Why is there a lot of “noise” over here?
Classwork
Sketch a graph and complete the table to find the limit (if it exists). x
-0.1
-0.01
-0.001
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.
Why is there a lot of “noise” over here?