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$1/\infty$ 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.

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  • $\begingroup$ Determinate or indeterminate forms doesn’t correspond to whether the form obeys traditional algebraic rules, just whether the limit is unambiguously defined. $\endgroup$ – eepperly16 Apr 25 at 5:14
  • $\begingroup$ I’m not sure where but this question has multiple duplicates $\endgroup$ – gen-z ready to perish Apr 25 at 5:42
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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 $\frac {a_n} {b_n} \to 0$

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  • $\begingroup$ As an example of the latter, consider $a_n=n^2$ and $b_n=n$ vs. $a_n=n$ and $b_ n=n^2$ $\endgroup$ – J. W. Tanner Apr 25 at 5:09
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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.

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  • $\begingroup$ It is not $0$. It tends to 0. $\endgroup$ – Paras Khosla Apr 25 at 7:52
  • $\begingroup$ when we say 'anything times infinity is infinity', since 0 is different from rest numbers. So we can't say yet zero times infinity is infinity because that is the question. But with 'anything times zero is zero', here anything, include 0 or any numbers. Hence , $\infty$ * 0= 0, since no conflict as in the former. $\endgroup$ – Rajesh Marndi Apr 25 at 12:57
  • $\begingroup$ @ParasKhosla I updated my answer. $\endgroup$ – Max Mir May 10 at 3:52
  • $\begingroup$ "Indeterminate forms" refer to something that occurs while you are trying to take a limit. I think it is impossible to say anything useful about indeterminate forms unless it is understood how limits are involved. Contrary to what you wrote, the quotient of a very large number divided by a very large number is perfectly well defined. Moreover, $\lim_{x\to0^+}f(x)/g(x)=3$ when $f(x)=1+3/x$ and $g(x)=2+1/x$ even though $\lim_{x\to0^+}f(x)=\infty$ and $\lim_{x\to0^+}g(x)=\infty$. But if you write $f(0)/g(0)$, $f(0)$ is not defined, so you don't even get $\infty/\infty.$ $\endgroup$ – David K Nov 14 at 13:57

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