# 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

necessarily1. 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.

**See Also**

**Cite as:**Is Infinity / Infinity = 1?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/is-fracinftyinfty1/