Indeterminate Forms Misunderstood

Question

Hello everyone my question may be stupid but I really don't understand why $$0 \times +\infty$$ is undefined since o times any number is zero no matter how large the number may be

Answer 1

The fact is, $+ \infty$ is not a number. The laws of multiplication hold for a number $x\in \mathbb C \supseteq \mathbb R \supseteq \mathbb Q \supseteq \mathbb Z \supseteq \mathbb N$, but infinity is not a natural number. If you start searching around your argument, you could observe that, for example :

$$\infty \times 0 = \lim_{x \to \infty} (x \times 1/x) = 1$$

$$\infty \times 0 = 0$$

But the above, are not statements that hold for infinity, because, as mentioned, it cannot be treated as a real/natural number.

Infinity must be defined, before working with and on it, but the problem is that there are many different ways to define it, exactly because it's not a normal number case.

For example, a different and classic case is that you could define infinity by $1/\infty = 0$ but in the same time, infinity can be defined while talking about infinite sets (cardinal numbers).

As it becomes clearer, you see that infinity is rather hard to define strictly and thus really hard to handle in such strict ways, as the multiplication law.

So, to sum up, any multiplication or summation law that holds for the natural numbers does not have any meaning for infinity and thus cannot be used to derive expressions and calculations, such as : $0 \times \infty = 0 $.

Also, take in account that an operation for a natural number $n$ such the one you mentioned, is :

$$n \times 0 = 0 + 0 + 0 +\cdots+ 0 = 0$$

which is clear that is a finite operation and not an operation regarding something infinite.

Answer 2

The easiest way of understanding this particular indeterminate form is as saying that an infinitesimal times an infinite number could have any desired order of magnitude: infinitesimal, appreciable, or infinite. Such a situation is summarized by the symbol $0\times\infty$ but in point of fact you are not actually multiplying zero by an infinite number but rather an infinitesimal by an infinite number.

For example, if $\epsilon>0$ is infinitesimal, then $H=\frac{1}{\epsilon}$ is infinite. Then the products $\epsilon^2\times H$, $\quad \epsilon\times H$, and $\epsilon\times H^2$ are respectively infinitesimal, appreciable, and infinite.

See this textbook for details.