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

[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)?

 

[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.

 

[micromass]

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

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.

 

[rfrederic]

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