Multivariable Limit: Does Not Exist

[Stevo6754]

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?

 

[HallsofIvy]

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 $(x^2+ 1- 2)/(x^2- 1)= (x^2- 1)/(x^2- 1)= 1$ and, with x= 1, $(1+ y^2- 2)(1- y^2)= (y^2- 1)/(1- y^2)= -1$.