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 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


  1. Example [Using Images] [Using Flash]


  2. Example [Using Images] [Using Flash]


  3. Example [Using Images] [Using Flash]
    Discussion [Using Flash]


  4. Example [Using Images] [Using Flash]
    Discussion [Using Flash]


  5. Example [Using Images] [Using Flash]
    Discussion [Using Flash]


  6. Example [Using Images] [Using Flash]


  7. Example [Using Images] [Using Flash]


  8. Example [Using Images] [Using Flash]


  9. Example [Using Images] [Using Flash]


  10. Example [Using Images] [Using Flash]

  11. provided when n is even.
    Example [Using Images] [Using Flash]

  12. If for all then .

  13. If f is a polynomial then .
    Example [Using Images] [Using Flash]
    Discussion [Using Flash]

  14. 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:

Theorem E. Suppose that f and g are two functions such that

and

then the limit

does not exist.