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.

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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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