Can you give an example to show why infinity times zero should not be defined?
$\displaystyle \lim_{x\to 0} ax= 0$ for all x and $\displaystyle \lim_{x\to 0}\frac{1}{x}= \infty$ but $\displaystyle \lim_{x\to 0}(ax)(\frac{1}{x})=\lim_{x\to 0}a= a$. If we defined "$\displaystyle 0(\infty)$" to be any specific value, the rule $\displaystyle \lim_{x\to a} f(x)g(x)= \left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a} g(x)\right)$ would not be true since the above examples would be of the form "$\displaystyle 0(\infty)$" but the limit of the product depends upon a.