Problems with x/∞

Question

If $\dfrac {x} {\infty }=0,$ where $x$ is a finite number, than wouldn't $0\cdot \infty $ be equal to any number? Making this not work?

Answer 1

The problem here is:

Division by $0$ or $\pm\infty$ is not generally defined!

It is not entirely mathematically correct to write $\dfrac{x}{\infty}=0$ when $x$ is real.

One thing you can do to mathematically justify your initial statement is to write it in the form of a limt:

$$\lim_{n\to\infty}\frac{x}{n}=0~\forall~x\in(-\infty,+\infty)$$

And then, you can write your claim (also in the form of a limit):

$$\lim_{n\to\infty}(0\times n)=0$$

Without using the form of limit, the values are undefined.