Show that the indicated limit does not exist for: lim┬((x,y)→(0,0))⁡〖(x^2+y^2)/(x^2+xy+y^2 )〗
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Show that the indicated limit does not exist for: lim┬((x,y)→(0,0))⁡〖(x^2+y^2)/(x^2+xy+y^2 )〗

[From: ] [author: ] [Date: 12-06-04] [Hit: ]
......
Fix m, and let y = mx:

lim(x→0) (x^2 + (mx)^2) / (x^2 + x * mx + (mx)^2)
= lim(x→0) x^2 (1 + m^2) / [x^2 (1 + m + m^2)]
= (1 + m^2)/(1 + m + m^2).

Since this dependent on the value of m, the multivariable limit does not exist.

I hope this helps!
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