Objectives: In this tutorial, we introduce an "intuitive" definition of a limit. After working through these materials, the student should be able

• to obtain numerical evidence for the calculation of limits;
• to determine what appears to be the limit from the numerical evidence; and
• to become aware of some of the problems in using numerical evidence for the calculation of limits.

Modules:

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

• Motivation for the definition and some examples.
  [Using Flash]

• Some more examples; these are randomly generated.
  [Using Flash] [Using HotEqn] [Using IBM's TechExplorer]

• An example involving the sine function.
  [Using Javascript]

• An example where the limit does not exist.
  [Using Javascript]

• An example where the left hand and right-hand limits exist but are not equal.
  [Using Javascript]

• Using a graphing calculator to obtain numerical approximations for limits:
  • TI-86 Graphing Calculator [Using Flash]
  • TI-85 Graphing Calculator

• Some examples that illustrate some problems with using numerical evidence for evaluating limits.
  [Using Javascript]