# Math Help - Products of limits

1. ## Products of limits

Can you give an example to show why infinity times zero should not be defined?

2. Originally Posted by Slazenger3
Can you give an example to show why infinity times zero should not be defined?
$\lim_{x\to 0} ax= 0$ for all x and $\lim_{x\to 0}\frac{1}{x}= \infty$ but $\lim_{x\to 0}(ax)(\frac{1}{x})=\lim_{x\to 0}a= a$. If we defined " $0(\infty)$" to be any specific value, the rule $\lim_{x\to a} f(x)g(x)= \left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a} g(x)\right)$ would not be true since the above examples would be of the form " $0(\infty)$" but the limit of the product depends upon a.