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Limits of a Rational Function of Two Variables


	What is the behavior of for positive and near , where , , , , and are positive? This Demonstration allows you to investigate.


	Contributed by: Roger B. Kirchner

, then

is bounded and the limit does not exist.

, then

is unbounded.

To see this, let

and

Then,

What about the limit of

along the algebraic curves

, where

, then

and

, then

can have limit 0, any positive number, or ∞, depending on

Let

and

be defined by

, and

Then

To see this, let

, so

, where

Then,

The results follow by considering the cases

, and

Example 1:

, as

, and along any curve to the origin.

Example 2:

, or

then

and

Example 3:

, or

, then

, and

Example 4:

Then,

and

, or

and

, then

Example 5:

Then

and

, or

, then

, and

"Limits of a Rational Function of Two Variables" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/LimitsOfARationalFunctionOfTwoVariables/

Contributed by: Roger B. Kirchner

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