1.2 Finding Limits Graphically and Numerically

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

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

5. Exploration p.48

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 • 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