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When I plug in values, I get -1/0 which is undefined. My question is, how do I prove that the limit does not exist? I can't set y=mx because y and x do not necessarily follow the same path to the point (1,-1,1) and using polar coordinate substitution does not seem to work here. I tried setting z=x but that still gives me a 0 in the denominator because of the -1 for the y. Finally, multiplying the equation by various values of 1 (such as multiplying y squared to both the numerator and denominator) don't solve the undefined issue.
Ah, if that were the case then the limit would be -1 ...
But then I have an undefined value and a real value ... does that mean the limit does not exist? Or do I have to keep trying different methods to get to that point and hope that those limits don't equal -1?
If the limit along one path to the limit point is different from the limit along a different path to the limit point, then the limit does not exist. On the other hand, if the limits along any path to the limit point are all the same, finite number, then the limit does exist.