Objectives: In this tutorial, we recall the definition of distance in the real line. We relate certain inequalities involving distance with open intervals. We start with the intuitive definition of a limit and express it in terms of distances. Combining these concepts together we obtain a geometrical interpretation of limits. After working through these materials, the student should be able

• to recognize geometrically whether a limit exists or not; and
• using technology, to find geometrically the d when a specific e is given.

Modules:

• Recall:

 Intuitive Definition. Let y = f(x) be a function. Suppose that a and L are numbers such that whenever x is close to a but not equal to a, f(x) is close to L; as x gets closer and closer to a but not equal to a, f(x) gets closer and closer to L; and suppose that f(x) can be made as close as we want to L by making x close to a but not equal to a. Then we say that the limit of f(x) as x approaches a is L and we write

• We refine this as follows:

 Definition. The limit of f(x) as x approaches a is L provided that choosing e > 0, indicating that we want the distance between f(x) and L to be less than e we can find d > 0 so that if the distance from x to a is less than d but not equal to 0 then the distance from f(x) to L will be less than e.

• Discussion of distance in the real line and the relationship between certain inequalities and open intervals.
[Using Flash]

• Motivation for the definition of limits using the intuitive definition and geometrical interpretation of limits.
[Using Flash]

• Examples in finding geometrically d when given a specific e.

• Examples using a graphing calculator to find geometrically d when given a specific e.

• An alternative way to use a graphing calculator to find geometrically d when given a specific e.

• Examples that illustrate limits that do not exist.

• An example of a limit that exists but the graphical evidence indicates that the limit does not exist.