why 1/ infinity isn't indeterminate like other indeterminate?

Question

$1/\infty$ tends to 0.

$\mathbf {It\ doesn't \ satisfy\ the\ inverse \ process\ of\ multiplication \ and \\division\ i.e} $

$\infty * 0$ is undefined or indeterminate.

So why $1/\infty$ is not indeterminate like other indeterminate $0/0$ , $\infty/\infty,..$

Thanks.

Answer 1

If $a_n \to 0$ and $b_n \to 0$ then $\frac {a_n} {b_n}$ may converge to any number or may not even converge. Same thing is true if $a_n \to \infty$ and $b_n \to \infty$. But if $a_n \to 1$ and $b_n \to \infty$ then $\frac {a_n} {b_n} \to 0$

Answer 2

1 over something very large ($\infty$) tends to 0. That much is clear. However, what does some very large number divided by another very large number even mean? Hence $\infty/\infty$ cannot be determined. Same with $0/0$ - it doesn't mean anything, hence indeterminate. You cannot determine what such a number is. Now, $\infty*0$ also is meaningless, since anything times infinity is infinity and anything times 0 is 0, so infinity times zero cannot be determined.