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An example where inverting infinities goes wrong is. Let ##f(x) = 2x## and ##g(x) = x##, then ##f(\infty) = \infty## and ##g(\infty) = \infty##, so:I am an engineer, and not a mathematician, so I understand I do not speak in the same precise terms as one. But I struggle to think of an infinity that can't be inverted, and, when it is, would not equal zero. Is there one?

##\frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1##

But

##h(x) = \frac{f(x)}{g(x)} = 2##

So ##h(\infty) = 2##

But ##h(\infty) = \frac{f(\infty)}{g(\infty)} = \frac{\infty}{\infty} = 1##

So, which is correct? Is ##\frac{\infty}{\infty} = 1## or is ##\frac{\infty}{\infty} = 2##?

One of the reasons you need to develop rigorous, well-defined mathematics is to resolve this sort of issue. And, one of the things you have to do is to stop using ##\infty## as a number.