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?

Numericallyx 0.75 0.90 0.99 0.999 1 1.001 1.01 1.10 1.25

f(x) ?

Notation

Lxfcx

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

Exploration p.48

2

232

2lim

x

xx

x

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

11lim0 x

x

x

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

2,0

2,1)(

x

xxf

1. Numerical Approach – Construct a table of values.2. Graphical Approach – Draw a graph by hand or using technology.3. Analytical Approach – Use algebra or calculus.

Limits that Fail to Exist

x

x

xlim

0

1. Numerical Approach – Construct a table of values.2. Graphical Approach – Draw a graph by hand or using technology.3. Analytical Approach – Use algebra or calculus.

Ex 4: Unbounded Behavior

20

1lim xx

1. Numerical Approach – Construct a table of values.2. Graphical Approach – Draw a graph by hand or using technology.3. Analytical Approach – Use algebra or calculus.

Ex 5: Oscillating Behavior

xx

1sinlim

0

1. Numerical Approach – Construct a table of values.2. Graphical Approach – Draw a graph by hand or using technology.3. Analytical Approach – Use algebra or calculus.

x 2/π 2/3π 2/5π 2/7π 2/9π 2/11π As x approaches 0?

x

1sin

Common Types of Behavior Associated with the Nonexistence of a Limit

1. f(x) approaches a different number from the right side of c than it approaches from the left side.

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

Assignment: Section 1.2a

• Section 1.2 (2 – 20)even