Rule for calculating big-O plus example I can't figure out I have the following rule:
If $f$ is $O(g)$ for ${x\to\infty}$ and $\lim_{x\to\infty}g(x) = 0$ than also $\lim_{x\to\infty}f(x) = 0$
Then my text proceeds to give an example:
$\lim_{x\to\infty}\frac{3^x+x^2+x\sqrt{x^5}-\ln(x^2+1)}{2\cdot3^{x+1}-2^x+\sqrt{1+x^6}}$
And it says: "We will use the fact that the last three terms of the numerator are $O(x^4)$, and the fact that the last two terms of the denominator are $O(2^x)$". This is what happens next:
$\lim_{x\to\infty}\frac{3^x+O(x^4)}{2\cdot3^{x+1}+O(2^x)} = \lim_{x\to\infty}\frac{1+O(\frac{x^4}{3^x})}{2\cdot3+O(\frac{2^x}{3^x})} = \frac{1}{6}$
I fail to see how the rule is used, because e.g. if $O(g)$ is $O(x^4)$, this does not equal zero. I also can't figure out what $f$ is supposed to be.

Here $f$ is $\frac{x^2+x^{\frac{7}{2}}- \ln (x^2+1)}{3^x}$ and $g$ is $\frac{x^4}{3^x}$ for the numerator. Why the system ask me whether i'm a human being?
