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Disproving Multivariable Limit

I just started learning about multivariable functions, and I was wondering if there is something like Heine for disproving the limit of multivariable functions.

For example: I know that we can approach multivariable functions from different directions to see if the function keep the same value, but is this enough? Is there a more rigorous way to do it? Like Heine was used in one variable functions?

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