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From what I understand, evaluating limits of multi-variable functions (for simplicity let's say 2 variables) means that you have to eliminate one of the variables by finding a "path" on which to approach the limit. The way I understand it is that if I'm trying to evaluate $$ \lim_{(x,y) \rightarrow (0,0)}\frac{x^2y}{x^4+5y^2}$$ I can say that $y=x^2$ is my "path" and then get $$ \lim_{x \rightarrow 0}\frac{x^4}{6x^4}=\frac{1}{6}$$ But just because using $y=x^2$ gave me $\frac{1}{6}$ that doesn't necessarily mean all paths do. So if I want to approach (0,0) from all directions, could I take my path to be something like an Euler spiral/Cornu spiral/clothoid centered on (0,0), which would circle the point infinitely as it approached it, eventually converging on the point from all directions?

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  • $\begingroup$ Yes, you can take any path that converges to the desired limit, including an Euler spiral. However, you will still have the same problem: just because using a specific spiral gives you a certain limit doesn't necessarily mean all paths do! For example, one can construct a function that is 0 for all points on the spiral and is 1 elsewhere. $\endgroup$ – Rahul Apr 3 '18 at 5:55
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This idea with paths or sequences $(x_n,y_n)$ tending to $(0,0)$ can only be used to prove that some limit does not exist: You obtain a particular counterexample, which then nullifies the conjecture that we have a limit.

It is true that the limit $\lim_{(x,y)\to(0,0)} f(x,y)$ exists iff for all paths converging to $(0,0)$ we obtain the same limit, but you cannot use this fact for a limit proof in practice: There are more paths converging to $(0,0)$ than there are atoms in the universe, and to test all of them would take the first second of eternity.

In fact, if you cannot verify the existence of the limit from simple principles, you have to provide a full $\epsilon/\delta$-proof.

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