Let $f,g:[0,\infty)\to [0,\infty)$ be continuous function, and recall that $f=O(g)$ if there exists $C>0$ such that for sufficiently large $x$, we have $f(x)\leq Cg(x)$, and define $g=O(f)$ similarly. I found a statement that says
There exists $L\in \mathbb{R}, |L|<\infty$ such that $\lim_{x\to\infty}f(x)/g(x)=L$ if and only if $f=O(g)$ and $g=O(f)$.
Is this statement correct? It is easy to see that $\lim_{x\to\infty}f(x)/g(x)=L$ implies $f=O(g)$ and $g=O(f)$. What about the other implication? Is it wrong or is it provable? If provable, I would appreciate any comments on how to prive it.
$f=2+\cos(x)$ and $g=2+\sin(x)$.
I always have $f<4g$ and $g<4f$.
However $f/g$ does not converge