Objectives: The following is a list of theorems that can be used to evaluate many limits. After working through these materials, the student should know these basic theorems and how to apply them to evaluate limits.

Modules:

 Theorem A. Suppose that f and g are functions such that f(x) = g(x) for all x in some open interval interval containing a except possibly for a, then • Discussion of Theorem A [Using Flash]

 Theorem B. Suppose that f and g are functions such that the two limits exist, suppose that k is a constant and suppose that n is a positive integer. Then Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] Discussion [Using Flash] Example [Using Images] [Using Flash] Discussion [Using Flash] Example [Using Images] [Using Flash] Discussion [Using Flash] Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] Example [Using Images] [Using Flash] provided when n is even. Example [Using Images] [Using Flash] If for all then . If f is a polynomial then . Example [Using Images] [Using Flash] Discussion [Using Flash] If f is a rational function then, for all a in the domain of f, . Example [Using Images] [Using Flash] Discussion [Using Flash]

 Theorem C. The limit if and only if the right-hand limits and left-hand limits exist and are equal to M: • Examples:

1. where [Using Flash]

2. [Using Flash]

 Theorem D. (Squeeze Theorem) Suppose that f, g and h are three functions such that f(x) < g(x) < h(x) for all x. If then • Example:

1. [Using Flash]

 Theorem E. Suppose that f and g are two functions such that and then the limit does not exist.

• Example:

1. [Using Flash]