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Homework Statement
When we are taking a limit of a multivariable function, why do we use the points (x,0) and (0,y) to find out if the limit doesn't exist? They are two different points, no? If they are two different points then wouldn't they go to different points on the z-axis?
If we have a function, f(x,y), the output will be on the z-axis.
There are infinitely many ways in which we can get to a point on the surface (z-axis)
So if we start at the point (x,0) and apply the function we should end up at a point z.
and if we start at the point (0,y) and apply the function we should end up at the same point z.
Is that statement correct? It doesn't sound right. I'm having a little conceptual problem here so I'm trying to understand it in very simplistic language before i put it into more mathematically rigorous terms.
If we want to get to point C (f(x,y)), we can get there from point A (0,y) or point B (x,0). We would have to take different paths to get to C, but we WOULD get to C. If we found the C was different for A and B then we would say C (the limit) does not exist.
But then if there are infinitely many paths I can take to get to C, then doesn't that mean there are infinitely many points on the z-axis that I could go to? If that is so, then how do we know (0,y) and (x,0) should end up at the same point?
Sorry, I hope I haven't confused anyone with my question.