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I've only recently begun studying (in math, anyway) functions of two variables, so forgive me if my terminology is vague or incorrect (or if I am completely misguided!).

Suppose I want to find the limit of f(x,y) as (x,y) --> (0,0). Now, I know that the limit will exist (other conditions satisfied) only if f(x,y) --> A as (x,y) --> (0,0) from ALL directions.

To prove that the limit = Q, would it thus be sufficient to be able to show a.) that for approaches along ALL lines y = cx (c some real number), the limiting value f(x,y) will be independent of the value of c? (and since all curves through the origin can be approximated in this vicinity by a tangent line through the origin, surely this implies that all possible directions are accounted for) And then b.) that if the limiting value can be found to be Q for approach along SOME line, then by a.), the limit must exist and = Q?

Is this acceptable? Thanks for your help.

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# Homework Help: Limits: Fxns of 2 variables

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