1. 1.2 Finding Limits Graphically and Numerically
An Introduction to Limits
Limits that Fail to Exist
A Formal Definition of a Limit

2. An Introduction to Limits
Graph: f(x) = (x3 – 1)/(x – 1), x ≠ 1
What can we expect at x = 1?
Approach x=1 from the left.
Approach x=1 from the right.
Are we approaching a specific value from both sides? What is that number?

3. Numerically x
0.75
0.90
0.99
0.999 1
1.001
1.01
1.10
1.25
f(x) ?

4. Notation
The limit of f(x) as x approaches c is L.

5. Exploration p.48 x
1.75
1.90
1.99
1.999 2
2.001
2.01
2.10
2.25
f(x)

6. Example 1: Estimating a Limit Numerically
Where is it undefined?
What is the limit?

7. Continued Example 1: Estimating a Limit Numerically
It is important to realize that the existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c.

8. Ex 2: Finding the limit as x → 2
Numerical Approach – Construct a table of values.
Graphical Approach – Draw a graph by hand or using technology.
Analytical Approach – Use algebra or calculus.

9. Limits that Fail to Exist
Numerical Approach – Construct a table of values.
Graphical Approach – Draw a graph by hand or using technology.
Analytical Approach – Use algebra or calculus.

10. Ex 4: Unbounded Behavior
Numerical Approach – Construct a table of values.
Graphical Approach – Draw a graph by hand or using technology.
Analytical Approach – Use algebra or calculus.

11. Ex 5: Oscillating Behavior
2/π
2/3π
2/5π
2/7π
2/9π
2/11π
As x approaches 0?
Numerical Approach – Construct a table of values.
Graphical Approach – Draw a graph by hand or using technology.
Analytical Approach – Use algebra or calculus.

12. Common Types of Behavior Associated with the Nonexistence of a Limit
f(x) approaches a different number from the right side of c than it approaches from the left side.
f(x) increases or decreases without bound as x approaches c.
f(x) oscillates between two fixed values as x approaches c.

13. Assignment: Section 1.2a
Section 1.2 (2 – 20)even