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I just confused myself trying explain "infinity" to my daughter.

Infinity and a half is infinity. Check.

Half of infinity is infinity. Check.

Infinity minus half infinity is...what?

I get that $\infty-\infty$ is indeterminate, but Inf*(1-0.5) is obviously infinity. So I guess factoring out infinities is a no-go. That's fine. What about taking the limit?

$$\lim_{x\to\infty} x - 0.5x=\infty$$

Is that right? If it is, why is that not equivalent to $\infty-0.5\infty$?


marked as duplicate by user296602, JMoravitz, Henning Makholm, Community Jun 28 '18 at 1:09

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    $\begingroup$ There are a lot of other possible duplicates. The very short answer is that you can't take limits of individual pieces when you have an indeterminate form like this, and this has been handled in many questions here. $\endgroup$ – user296602 Jun 28 '18 at 0:57
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    $\begingroup$ The end result is that the question of "$\infty-\infty$" or "$\infty - 0.5\infty$" are illformed and unanswerable without clarifications. The question of "what is $\lim_{x\to\infty}(x-0.5x)$" however is perfectly fine and answerable. Indeed, most questions about "infinity" and arithmetic will be unanswerable until you give an appropriate related question about limits so one can know "which infinity" and how it relates to other infinites in the problem. $\endgroup$ – JMoravitz Jun 28 '18 at 0:59
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    $\begingroup$ Be careful in introducing infinity to anybody. Most people are not ready for it. And until you are ready to get rigorous with the definitions you might be better of saying that infinity is a concept and not a number and shouldn't be used in any algebraic statements without the proper framework. $\endgroup$ – Doug M Jun 28 '18 at 1:04
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    $\begingroup$ Some mathematicians prefer not to consider infinity as a number, rather they use it when there is a change without bound. For example, to say mathematically that the function $y=x$ grows larger and larger without bound, they use $y\to \infty$ as $x\to \infty$. But they do not consider expressions like $\infty - \infty$ since substraction and other similar operations are defined only for numbers $\endgroup$ – user555729 Jun 28 '18 at 1:09
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    $\begingroup$ The limit of $x-0.5x^2$ is also like $\infty-0.5\infty$, but is $-\infty$. The limit of $x-0.5(2x-c)$ is also like $\infty-0.5\infty$, but is the number $c$. So in this sort of sense it can be pretty much anything, which is why it's indeterminate. $\endgroup$ – Mark S. Jun 28 '18 at 1:13

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