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Division by $0$
What is the exact value of $1\over0$? In $y=$ $1\over{x}$, it shows it is $\infty$ and $-\infty$ (infinite and minus infinite).
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1It’s undefined. – Brian M. Scott Nov 10 '12 at 09:59
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1Amongst others: http://math.stackexchange.com/questions/71114/division-by-zero http://math.stackexchange.com/questions/144715/will-division-by-zero-be-defined-eventually http://math.stackexchange.com/questions/87781/defining-division-by-zero http://math.stackexchange.com/questions/26445/division-by-0 http://math.stackexchange.com/questions/231774/limit-as-x-to-0-with-zero-division – Asaf Karagila Nov 10 '12 at 10:03
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Division by zero cannot be defined in a meaningful way.
If you interpret $1/0$ as a one-sided limit of $1/x$ (and accept improper limits), then
$$\lim_{x\to 0^+} \frac1x = +\infty \qquad\text{and}\qquad \lim_{x\to0^-} \frac1x = -\infty.$$
mrf
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Let $x \in \mathbb{R}^*$. By definition, $1/x$ is the unique real $y$ such that $xy=1$. So you see that you can't extend this definition to include $1/0$, because for all real $y$, $0y=0$.
Another possibility would be to extend $x \mapsto 1/x$ at $0$ by continuity, but there is no such extension as $\lim\limits_{x \to 0^{\pm}} \frac{1}{x} = \pm \infty$.
In fact, there is no really natural way to define $1/0$.
Seirios
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1@HagenvonEitzen: Sometimes, it is quite natural to identify $+\infty$ with $-\infty$. But only do it when necessary, and if you know what you are doing. The result of doing so is known as the projective line. – Harald Hanche-Olsen Nov 10 '12 at 10:28
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@HaraldHanche-Olsen: Yes, I know. My "Don't" was therefore intended for the level where the current question arose (I as in a kind of typing hurry and hope that your comment made things clearer) – Hagen von Eitzen Nov 10 '12 at 15:20
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@HagenvonEitzen: I expected as much, and my comment wasn't really directed at you – I only named you to let you know I commented. – Harald Hanche-Olsen Nov 10 '12 at 15:29
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