Why $x^2+3x^4+\omicron(x^4)=\omicron(x^3)$is false? Why $x^2+3x^4+\omicron(x^4)=\omicron(x^3)$is false for $x\to 0$?
I know that:
A. $-\omicron(f)=\omicron(f)$
B. $\omicron(f)+\omicron(f)=\omicron(f)$
C. If g is a function $g*\omicron(f)=\omicron(gf)$
C2. if g is costant$g*\omicron(f)=\omicron(gf)=\omicron(f)$
D. $\omicron(f)^n=\omicron(f^n)$
E. $\omicron(f+\omicron(f))=\omicron(f)$
From C I try to write $\omicron(x^4)=x^4\omicron(1)$ but I don't know what my next move is.
If
$$x^2+3x^4+\omicron(x^4)=\omicron(x^3)$$ then
$$\lim\limits_{x \to 0} \frac{x^2+3x^4+\epsilon(x)x^4}{x^3} = 0$$
where $\lim\limits_{x \to 0} \epsilon(x) =0$.
And this is not the case as
$$\frac{x^2}{x^3}$$ is not bounded around zero.
Dividing by $x^2$,
$$1+3x^2+o(x^2)=o(x)\ ???$$