Introduction to Limits

Objectives: The first part of this tutorial contains a list of theorems that can be used to evaluate many limits. The second part contains a collection of examples that these theorems cannot be used to evaluate immediately. It is shown how to do some algebraic manipulation to put these examples in the form so that the theorems can be applied. After working through these materials, the student should know these basic theorems, how to apply them to evaluate limits and how to manipulate certain examples so that the theorems may be used.

Modules:

To help us in the symbolic or algebraic computation of limits, we have a list of limit theorems.
[Using Flash] [Using HotEqn] [Using IBM Professional TechExplorer]
As a result of these theorems, we see that for many functions f,

A function which has this property is called continuous. From the above-mentioned list of limit theorems, we see that polynomial functions and rational functions are continuous. We will study continuous functions more extensively in another module.
The following examples demonstrate how we can evaluate limits of functions which are not continuous by using the above-mentioned list of limit theorems. These include many of the examples which were explored numerically and/or graphically.
Examples.
- Discussion [Using Flash]
- Discussion [Using Flash]
- Discussion [Using Flash]
- where
  
  Discussion [Using Flash]
- where
  
  Discussion [Using Flash]
- where
  
  Discussion [Using Flash]
- Discussion [Using Flash]
- Discussion [Using Flash]
A LiveMath notebook showing another application of the Squeeze Theorem.
Drill problems on evaluation of limits.
A quiz on using the limit theorems and graphs to evaluate limits.
Using a graphing calculator:
- An interesting example about limits and the greatest integer function.
  - TI-86 Graphing Calculator [Using Flash]
  - TI-85 Graphing Calculator
Using Maple: