Limits multivariable Problem?????

Question

I am not really sure how to solve this problem...

 

Evaluate the Limit:

 

Limit 

(x,y,z) ----> (0,0,0)            (x²+y²+z²)/(x²-y²-z²)

 

 

Answer 1

This limit does not exist.  

 

If you hold z constant at 0 and approach the origin along the line y = 2x, then 

f(x,y,z) = f(x,2x,0) = (x² + 4x² + 0) / (x² - 4x² - 0) = (5x²) / (-3x²) = -5/3 

so the limit as (x,y,z)--->(0,0,0) along this line is -5/3 

 

But if you approach the origin along the line y = 3x with (z = 0) then 

f(x,y,z) = f(x,3x,0) = (x² + 9x² + 0) / (x² - 9x² - 0) = (10x²) / (-8x²) = -5/4 

so the limit as (x,y,z)--->(0,0,0) along this line is -5/4 

 

In order for the limit to exist it must have the same value regardless of which path you use to approach the limit point.  Since these 2 paths approaching the origin have different limiting values the limit does not exist.