What is the exact value of $1\over0$? [duplicate]

Question

Possible Duplicate:
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).

Answer 1

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 $1/0$.

Answer 2

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

$$\lim_{x\to 0^+} \frac1x = +\infty \qquad\text{and}\qquad \lim_{x\to0^-} \frac1x = -\infty.$$