Is it possible to let limit multiplication rule not be exist?

1. Oct 8, 2012

rfrederic

We have known that:
lim f(x)g(x) = lim f(x) * lim g(x)

Could any conditions or limitations make: lim f(x)g(x) ≠ lim f(x) * lim g(x)?

2. Oct 8, 2012

MarneMath

Writing out the proof really quickly, I don't see what kind of limitation or conditions can make this not true. Perhaps if you proved it, you would see why this is always true?

Edit: Based on micro post, I did the terrible mistake of assuming you meant that the x approached the same a and both limits exist.

Last edited: Oct 8, 2012
3. Oct 8, 2012

micromass

Staff Emeritus
No, this is not known. And this is not even true. You need to be very careful with statements like this. The correct statement is that

$$\lim_{x\rightarrow a} f(x)g(x)=\lim_{x\rightarrow a} f(x) \lim_{x\rightarrow a} g(x)$$

if both limits $\lim_{x\rightarrow a} f(x)$and $\lim_{x\rightarrow a} g(x)$ exist and equal a real number.

If we allow the limits to become infinite, then the rule does not hold anymore. For example, it is not true that

$$\lim_{x\rightarrow 0} \frac{x^2}{x^2}=\lim_{x\rightarrow 0} x^2 \lim_{x\rightarrow 0}\frac{1}{x^2}$$

since the limit $\lim_{x\rightarrow 0}\frac{1}{x^2}$ is not a real number (but rather $+\infty$).

You need to be very careful in mathematics to always give the exact statements and conditions in which something holds.

4. Oct 8, 2012

rfrederic

Big Thanks to micromass and MarneMath, both of you bestead me a lot.