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

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?

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

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

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

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

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.

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.

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.

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.

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.

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.

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