[Multivariable] How to prove this limit does not exist

I'm glad you didn't get it with the y=mx method, because I think it demonstrates something so important.

For the limit of a function in R^2 (or R^n) to exist, the limit has to be the same along ALL paths, not just linear paths. When you substitute y=mx, you're approaching your point head-on in a linear path. If you get that the limit along all these linear paths is NOT the same, it implies the limit does not exist.

But if you get the limit along all these paths is not the same, it DOES NOT imply the limit exists, it just fails to imply it doesn't (so basically it does nothing).

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With that being said, you can actually do this just checking linear paths, try y=x and y=-x, they will give different values.

y=x:

2x/x^2 = 1/x which has lim = infinity

y=-x:

0/(-2x^2) = 0.

So along two different paths the function doesn't agree. (Even if it did, infinity implies the limit doesn't exist anyway, but in case you also wanted to show different paths give different results you could do the extra step I did).

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