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.
Let y = f(x) be a function. Suppose that a and L are numbers such that
Then we say that the limit of f(x) as x approaches a is L and we write
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.
- Motivation for the definition and some examples.
- Some more examples; these are randomly generated.
[Using IBM's TechExplorer]
- An example involving the sine function.
- An example where the limit does not exist.
- An example where the left hand and right-hand limits exist but are
Using a graphing calculator to obtain numerical approximations for limits:
- Some examples that illustrate some problems with using numerical evidence for evaluating limits.