Products of limits

**Slazenger3** · February 4th 2010, 07:17 PM

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

**HallsofIvy** · February 5th 2010, 03:56 AM

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.

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