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 $\color{#D61F06}{\textbf{false}}$.

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.

See Also

• List of Common Misconceptions