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 \( \color{red}{\textbf{false}}\).
Proof: We know we cannot do arithmetic with infinity. But let's take a limit and see if it is true:
\[\lim_{x\to\infty} f(x)=\infty,\quad \lim_{x\to\infty} g(x)=0,\quad \lim_{x\to\infty} f(x)g(x)=\,?\]
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.
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.
\[\begin{align} f(x)=x,\ \ g(x)&=\dfrac{2}{x^3}\\\\\\ \lim_{x\to \infty} f(x)g(x)&=\, ? \end{align}\]