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After looking at this answer Problem with multivariable calculus: $\lim_{(x,y)\to (0,0)} \frac{x^3 + y^3}{x^2 + y}$ I wondered if you have a limit $$\lim_{(x,y)\to(0,0)}f(x,y)$$ And you found paths such that the limit is equal to something $0$ in this case but you take a path $y=g(x),\lim_{x\to0}g(x)=0$ and you have that $$\lim_{(x,y)\to(0^+,0^-)}f(x,g(x))=-\infty,\lim_{(x,y)\to(0^-,0^+)}f(x,g(x))=\infty$$ Does that imply the limit doesn't exist or the path we take must have two-sided limit when approaching $(0,0)$

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kingW3
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The limit as $(x,y)$ goes to $(0,0)$ exists if and only if the limit along any path exists and is the same. In the case you describe the limit does not exist.

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    @Emin Why does the limit need to be equal to the function value at the limit point in order for the limit to exist? (It needs to be equal in order to say that the function is continuous, but nobody was asking about continuity.) – David K Sep 19 '15 at 12:39