Explanation of method for showing that $\frac{0}{0}$ is undefined (This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is undefined:

$x = \frac00$
   $x \cdot 0 = 0$  
Hence There are an infinite number of values for $x$ as anything multiplied by $0$ is $0$.

However, it seems to have got comments, with two general themes.
Once is that you lose the values of $x$ by multiplying by $0$.
The other is that the last line is: 

$x \cdot 0 = \frac00 \cdot 0$

as it involves a division by $0$.
Is there any merit to either argument? More to the point, are there any major flaws in my explanation and is there a better way of showing why $\frac00$ is undefined?

Ok, here is one reason why dividing by $0$ has no meaning...
Dividing by $0$ would mean to multiply by the inverse of $0$.
However, the inverse of $x$ is the number $y$ for which $x \times y=1$.
Since $\forall x \in \mathbb{R}, 0 \times x=0$, $0$ does not have any inverse. 
Hence, you can't divide by $0$.
