Thales was the first known Greek philosopher, scientist and mathematician. He is credited with five theorems of elementary geometry. Thales was the first known Greek philosopher, scientist and mathematician. He is credited with five theorems of elementary geometry. Thales of Miletus 625 – 547 B.C. Thales of Miletus 625 – 547 B.C. The Ancients

As x approaches 2, what is the behavior of f (x)? There will be three strategies for analyzing this question. 1.) Graphically 2.) Numerically 3.) Algebraically/Symbolically In section 2.2, our analysis will be graphical and numerical, the warm and fuzzy way! Example 1

The limit of a function refers to the value that the function approaches, not the actual value (if any)

Solution Example 2 Hole in the graph

Properties of Limits: For a limit to exist, the function must approach the same value from both sides. One-sided limits or directional limits approach from either the left or right side only. Example 3

Properties of Limits: For a limit to exist, the function must approach the same value from both sides. One-sided limits or directional limits approach from either the left or right side only. Example 3

So, lets consider the function f (x) below. Limit from the left Limit from the right These are called one-sided or directional limits. These are called one-sided or directional limits. From the left From the right

So, lets consider the function f (x) below.

At x = 1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match! Example 4

At x = 2 : left hand limit right hand limit value of the function because the left and right hand limits match Example 4

At x = 3:left hand limit right hand limit value of the function because the left and right hand limits match All three are equal! I wonder if that is something special? Example 4

Example 5

Example 6

= 4 DNE Example 7 Find each limit. Quick Quiz

Example 8

Example 9 Example 10

Example 11

Example 12