calculus - Why is $infty cdot 0$ an indeterminate form, if $infty$ can be treated as a very large positive number?

Why is $\infty \cdot 0$ indeterminate?

Although $\infty$ is not a real number, it can be treated as a very large positive number, and any number multiplied by $0$ is $0$. Why not in this case?

Answer

Why is $\infty \cdot 0$ indeterminate?

It's called indeterminate because if you get $\infty \cdot 0$ when evaluating a limit, then you can't conclude anything about the result.

Here's an example:

$$\lim_{n \to \infty} (n^2) \cdot \left(\frac{1}{n}\right).$$
Now, this limit is $\infty$. But the limit of the first thing $(n^2)$ is $\infty$, and the limit of the second $(1/n)$ is $0$. So we cannot evaluate $\infty \cdot 0$ to compute the limit.

More technically: If we have one sequence $a_1, a_2, a_3, \ldots$ of real numbers that approaches $\infty$, and another $b_1, b_2, b_3, \ldots$ that approaches $0$, the product sequence $a_1 b_1, a_2 b_2, a_3 b_3, \ldots$ might approach any real number, or it might not approach anything at all.
This is what it means for $\infty \cdot 0$ to be an "indeterminate form".

Although $\infty$ is not a real number, it can be treated as a very large positive number

No, I would not agree with this statement.
It may be helpful to intuitively think of $\infty$ as a very large positive number, but this is not what infinity is. $\infty$ is a sort of limit of higher and higher positive numbers.
In the same way $0$ is a limit of lower and lower positive numbers.

and any number multiplied by $0$ is $0$. Why not in this case?

As explained above, $\infty$ is not a very large number, but rather a limit of larger and larger numbers. So we cannot say that multiplying $\infty$ by $0$ is $0$.