Limits of a Rational Function of Two Variables

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

Contributed by: Roger B. Kirchner (March 2011)
Open content licensed under CC BY-NC-SA


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Details

If , then as .

If , then is bounded and the limit does not exist.

If , then is unbounded.

To see this, let and .

Then, .

What about the limit of along the algebraic curves , where ?

If , then .

If , then

if and

if .

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

Let and be defined by

, and

.

Then if or ,

if ,

if ,

if or .

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:

,

If , or , ,

then if and if .

Example 3:

,

If , or , , then

if , and if .

Example 4:

,

Then, and .

If , or and , then

if or ,

if ,

if ,

if .

Example 5:

,

Then and

If , or , , then

if or ,

if ,

if , and

if .



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