Is Infinity / Infinity = 1?
This is part of a series on common misconceptions.
Is this true or false?
\[\dfrac{\infty}{\infty}=1\]
Why some people say it's true: Any number divided by itself is 1.
Why some people say it's false: We cannot just do arithmetic with something that is not a number.
The statement is \( \color{red}{\textbf{false}}\).
Proof: We know we cannot do arithmetic with infinity. But let's take a limit and see if it is true:
\[\lim_{x\to\infty} f(x)=\infty,\quad \lim_{x\to\infty} g(x)=\infty,\quad \lim_{x\to\infty} \dfrac{f(x)}{g(x)}=?\]
We know two such functions are \(f(x)=2x\) and \(g(x)=x\). But the limit is then 2 and not 1, and hence it is not necessarily 1. The limit is multivalued and \(\frac{\infty}{\infty}\) is undefined. \(_\square\)
Rebuttal: If \(\frac{\infty}{\infty}\neq 1\), then \(\infty\neq \infty\).Reply: You are cross multiplying, but it is not legitimate here. Let's multiply both sides with \(\ \infty\). We get \(\infty\times\frac{\infty}{\infty}\neq 1\times\infty\). Then you assumed that the infinities would cancel out to one, but remember they are not 1.
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