Indeterminate vs Infinite

Question

In mathematics, can the terms 'Infinite' and 'Indeterminate' be used interchangeably?

For example,

Can I say that $\frac{0}{0}$ is indeterminate/infinite?

No. For example, the limit of $\frac{x}{x}$ as $x \to 0$ is 1, while $\frac{x^n}{x} = 0$ for $n>1$. It is not infinite, just indeterminate. — Weaam, Jul 04 '17 at 05:56
@Weaam are you talking about the limit being 0 with $\frac{x^n}{x} = 0$ as x approaches 0? It isn't exactly clear. — rb612, Jul 04 '17 at 06:01
@rb612 Yes. Apply L'hopital's rule. It is $\lim_{x \to 0} x^{n-1} = 0$. — Weaam, Jul 04 '17 at 06:08
@R004 As $x\to 0$, we have the numerator $f \to 0$ and denumerator $g \to 0$ but we can't decide whether $f/g$ is $1$ or $0$ (or others), since if $f = x$, $g = x$ it is 1, while $f = x^n, n> 0$, we have $f/g \to 0$, hence the form $f/g$, which is $0/0$ in the limit, is indeterminate. — Weaam, Jul 04 '17 at 06:09
@R004 $;\frac{0}{0},$ is undefined. That's neither indeterminate nor infinite, but rather it has no meaning attached to it unless you very specifically define one. — dxiv, Jul 04 '17 at 06:19
How would you differentiate between indeterminate and undefined? — R004, Jul 04 '17 at 06:37
@R004 If you ask whether something is blue or red, you need to define what that something is, first. b~#xu? is neither blue, nor red, it is undefined unless you provide a definition for it. Same for $,\frac{0}{0},$. — dxiv, Jul 04 '17 at 06:43

score 2 · Accepted Answer · answered Jul 04 '17 at 06:11

2

No, they are different. Indeterminate means that it can "take on" multiple values. For example $\frac{0}{0}$ could be any number, or "infinity". It cannot be determined.

I think a more rigorous way to show that $\frac{0}{0}$ is indeterminate is by understanding that:

$$\lim_{x \to 0} \frac{x^2}{x} \neq \lim_{x \to 0} \frac{x}{x}$$

answered Jul 04 '17 at 06:11

rb612

3,560

If I understand well, you say that both $\frac{x}{x}$ and $\frac{x^2}{x}$ tend to $\frac{0}{0}$ as $x$ tends to zero, but the former takes the value $1$ and the latter takes the value $0$. Since this is the case, the situation is indeterminate, is it not? – R004 Jul 04 '17 at 08:03
Might I be clear. Tend to the 'form' $\frac{0}{0}$. – R004 Jul 04 '17 at 08:28
1

@R004 that's exactly right! – rb612 Jul 04 '17 at 20:48
One other doubt here. Am I right when I say that 'undefined' and 'indeterminate' are not the same? For example, the derivatives of a function $f$, continuous in $[a, b]$, at its end points are indeterminate while the derivative of a function $g$, continuous in $(a, b)$, at $a$ and $b$ are undefined. – R004 Jul 05 '17 at 11:27
1

@R004 you're right! Undefined does not have a value, indeterminate means its value cannot be determined precisely. – rb612 Jul 05 '17 at 17:49

score 2 · Answer 2 · answered Jul 04 '17 at 06:15

Indeterminate means 'impossible to calculate an exact value'. For example a system of $N$ equations and $M$ variables with $M\gt N$ is indeterminate because every variable can assume more than one value. Infinite means a set of elements with cardinality of $\mathbb{N}$ or $\mathfrak{C}$.

score 1 · Answer 3 · answered Jul 04 '17 at 06:16

No!

$\frac{0}{0}$ is indeterminate because we can't meaningfully "assign" any single value, you have already be shown various limits that shows how multiple values could be "sane".

E.g. $\lim_{x\to\infty} x^2$ is infinite because it grows beyond any limit, but it's not indeterminate because there's only one "value" that makes sense.

Indeterminate vs Infinite

3 Answers3