All possible paths to evaluate a multi variable limit

Most of the books that I have (H.K Dass) say that (or at least that’s what I have understood) for the limit of a multivariable function (say f(x,y) ) to exist the limit along every possible path should exist and be equal. That’s the condition for continuity.But for every solved example they do they just consider the limit along y=mx and y=mx^2 and if the limits in these 2 cases come equal they say the limit exists and is equal to answer that they get when evaluating the limit at the point by moving along the above given curves.But we should evaluate the limit along say y=mx^3 and y=mx^4 and say on. In fact in some questions (in which the degree of denominator is greater) in which they get the limit by just evaluating across the linear and quadratic path doesn’t come the same when we evaluate along cubic path. So how can they be sure by just evaluating along the linear and quadratic path. One should be checking for other paths too.But how far can one go checking all the paths. If I have got the concept wrong then what’s the correct one. An example would be appreciated

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