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I have the sense that f(x0,y0, ...) is indeterminate if the limit can be any complex number if we choose the right path to (x0, y0, ....). MathWorld and Wikedidia mention the subject, but it wasn't clear enough for me.

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2 Answers 2

My impression is that "indeterminate" is not a technical word with a precise meaning, but just a short way to say "undefined, and by the way be careful with taking limits here, because they won't behave sensibly".

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The term "indeterminate" is not really formally defined; it is used as a sort of vestigial phrase to describe certain limits in calculus (and, by extension, complex variables).

What it refers to is this. If you take a limit such at $\lim_{x \to 0} \frac{2x}{x}$ and replace the $x$ in the numerator and denominator with $0$, you get the fraction $0/0$. But the same thing happens if you take $\lim_{x \to 0} \frac{3x}{x}$ or $\lim_{x \to 0}\frac{0}{x}$, you also get $0/0$. But these three limits all have different values. We say that $0/0$ is the "indeterminate form" for these limits. There are ways to handle such limits, including L'Hoptial's rule, so it's useful to have a phrase for them.

The odd thing about this terminology is that if we simplify the expression, a limit that used to be indeterminate may no longer be. For example $\lim_{x \to 0} \frac{0}{x}$ and $\lim_{x \to 0} 0$ are both limits of the same function, $f(x) = 0$, as $x$ approaches the same point, $0$. But the first one is indeterminate and the second isn't. This is one reason why "indeterminate" is not defined in calculus books, in practice: because it depends not just on the function but on how it is written.

Separately: if $f(z)$ is a complex function defined except at a point $w$, and you can make the limit $\lim_{z \to w} f(z)$ have any value you like by picking the right path to $w$, then the function $f$ is said to have an essential singularity at $w$. The term "indeterminate" is not usually used for that.

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