Multivariable Limit: Does Not Exist

In summary, the limit of (x^2 + y^2 - 2) / (x^2 - y^2) as (x,y) approaches (1,1) does not exist. This is because the approach along the x and y axes does not reach the point (1,1). However, approaching along the lines y=1 and x=1 gives different results, leading to the conclusion that the limit does not exist.
  • #1
Stevo6754
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0

Homework Statement


Lim (x,y)->(1,1) of (x^2 + y^2 - 2) / (x^2 - y^2)


Homework Equations


None


The Attempt at a Solution


not continuous..

so I thought I would approach 1 from both x and y axises

lim x->1 (x^2 - 2)/(x^2) = -2
limt y->1 (y^2 - 2)/(-y^2) = 2

Does not exist right? Am I going about this the correct way?
 
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  • #2
No, you are not looking at it properly- you can't get to (1, 1) along the x or y axes! On the x-axis, y= 0 so, as x goes to 1, you are going to (1, 0), not (1, 1). Similarly, on the y-axis, x= 0 so, as y goes to 1, you are going to (0, 1), not (1, 1).

You could, instead, try approaching (1, 1) along the line y= 1 and then along the line x= 1. Now, with y= 1, the function becomes [itex](x^2+ 1- 2)/(x^2- 1)= (x^2- 1)/(x^2- 1)= 1[/itex] and, with x= 1, [itex](1+ y^2- 2)(1- y^2)= (y^2- 1)/(1- y^2)= -1[/itex].
 

1) What is a multivariable limit?

A multivariable limit is the value that a function approaches when its input variables approach a particular point. It is used to describe the behavior of a function at a specific point or along a specific path in a multi-dimensional space.

2) How is a multivariable limit different from a single variable limit?

A single variable limit only considers the behavior of a function as its input variable approaches a particular value. A multivariable limit, on the other hand, takes into account the behavior of a function as multiple input variables approach a point simultaneously.

3) What does it mean when a multivariable limit does not exist?

When a multivariable limit does not exist, it means that the function does not have a unique limit as its input variables approach a certain point. This can happen when the function approaches different values along different paths towards the point or when the limit approaches infinity.

4) What are some common reasons for a multivariable limit to not exist?

One common reason for a multivariable limit to not exist is when the function has a jump or discontinuity at the point being approached. Another reason is when the limit approaches different values along different paths due to oscillations or irregular behavior of the function.

5) How can I determine if a multivariable limit does not exist using a graph or table?

If a function has a jump or discontinuity at the point being approached, it will be evident in the graph or table. Additionally, if the limit approaches different values along different paths, there will be a noticeable difference in the values as the input variables approach the point from different directions.

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