Could $\frac x0 = \pm\infty$?

Question

Possible Duplicate:
Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?

I remember that dividing by zero is frowned upon, because it is said that there is no real answer. With the concept of limits, going from the negative direction to zero would give $-\infty$, and going towards zero from the positive direction would give $+\infty$. This is partially the reason that $\frac x0 = $ undefined, even with using limits.

But could $\frac x0$ be equal to $\pm\infty$? I suspect this is not the case, so please explain why this is incorrect.

Answer 1

Nothing can equal infinity since its not a real number.

You should read this http://answers.yahoo.com/question/index?qid=20090114213310AAsN4jB

Answer 2

Defining dividing by zero is much more trouble than it's worth. For example, we would expect that $\frac{x}{0}\cdot 0=x$, the same way as $\frac{x}{y}\cdot y=x$ for all other $y$, but upon dividing by zero we forget all about the $x$, thus $\pm\infty=\pm\infty\Rightarrow\frac{1}{0}=\frac{2}{0}\Rightarrow 1=2$, no good.

Sometimes, however, one might work with $\mathbb{C}\cup\{\infty\}$ (see http://en.wikipedia.org/wiki/Riemann_sphere), but then you give up some very useful niceties of working with a field.

Answer 3

I do not entirely agree with the answers posted so far.

First, a comment on something in the question: One should not write $\dfrac x0 =\text{undefined}$. Rather, one should say that the value of the expression $\dfrac x0$ is undefined. This is not that "is" of equality; this is the "is" of predication.

In some contexts, it makes sense to put a single $\infty$ at both ends of the real line $\mathbb R$, so that $\mathbb R \cup \{\infty\}$ is topologically a circle. That makes sense when dealing with either rational functions or trigonometric functions. It makes rational functions defined and continuous everywhere on $\mathbb R\cup\{\infty\}$ and it makes trigonometric functions defined and continuous everywhere on $\mathbb R$.