Is infinity times zero = zero?

This is part of a series on common misconceptions.

Is this true or false?

$$0\times \infty=0$$

Why some people say it's true: Zero times anything is zero.

Why some people say it's false: We cannot do arithmetic with infinity.

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:

$$\lim_{x\to\infty} f(x)=\infty,\quad \lim_{x\to\infty} g(x)=0,\quad \lim_{x\to\infty} f(x)g(x)=\,?$$

We know two such function are $f(x)=x$ and $g(x)=\frac{1}{x}$. But the limit is then $1$ and not 0, and hence it is not necessarily 0. We could make this limit any value that we wish, which is why $\infty\times 0$ is undefined. $_\square$

Rebuttal: But any number multiplied with zero is zero, why is this not the case here?

Reply: You are correct, but infinity is not a number. And hence it does not apply to infinity.

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Rebuttal: If $\infty \times 0 \neq 0$, then $0\neq 0$.

Reply: You are dividing by infinity, which is not legal here. We are assuming $\frac{\infty}{\infty}$ is defined, which has been disproven using a similar technique used in the problem.

• List of Common Misconceptions