Limits Theorems

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

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provided when n is even.
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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, .
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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
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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]