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lim (x,y)->(a,b) f(x) * g(y) = lim x->a f(x) * lim y->b g(x) (As well as the n dimensional case)

Also, why when you try to show that a limit doesnt exist you can keep a variable constant, or do something like y=x, or approach from some other path and show that one path of approach doesnt give the same limit as another path of approach. This statement seems kind of ambiguous to me. Is there a rigorous way of proving this using the definition of the limit in Rn?

These are'nt specific questions of the textbook just things i am curious about and want to gain a better understanding