I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.

I need a sensible definition of asymptotic equivalence when there is not necessarily an $x_0$ such that $f$ and $g$ are non-vanishing for all $x>x_0$.

That is, I need an analogue of the statement (which only makes sense if such an $x_0$ exists):

There exists $A>0$ and $B<\infty$ such that $$A\leq \left|\frac{f(x)}{g(x)}\right|\leq B$$ for all sufficiently large $x$. Equivalently, $f(x)=O(g(x))$ and $g(x)=O(f(x))$ as $x\rightarrow\infty$.

By ``sensible'', I require that the analogous statement firstly implies the above whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$ and, secondly, that it does not imply the stronger statement $|f(x)/g(x)|\rightarrow C$ as $x\rightarrow \infty$ whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$.

The application I have in mind is number theoretic. Specifically, I am interested in the relative asymptotics of $L(x)$ and $M(x)$.

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.

I need a sensible definition of asymptotic equivalence when there is not necessarily an $x_0$ such that $f$ and $g$ are non-vanishing for all $x>x_0$.

That is, I need an analogue of the statement (which only makes sense if such an $x_0$ exists):

There exists $A>0$ and $B<\infty$ such that $$A\leq \left|\frac{f(x)}{g(x)}\right|\leq B$$ for all sufficiently large $x$. Equivalently, $f(x)=O(g(x))$ and $g(x)=O(f(x))$ as $x\rightarrow\infty$.

By ``sensible'', I require that the analogous statement firstly implies the above whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$ and, secondly, that it does not imply the stronger statement $|f(x)/g(x)|\rightarrow C$ as $x\rightarrow \infty$ whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$.

EDIT: I also require that one does not need to know anything about the zeros of the pair, i.e. excluding particular subsets of the domain is not a satisfactory extension of the definition.

The application I have in mind is number theoretic. Specifically, I am interested in the relative asymptotics of $L(x)$ and $M(x)$.

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.

I need a sensible definition of asymptotic equivalence when there is not necessarily an $x_0$ such that $f$ and $g$ are non-vanishing for all sufficiently large $x$.

That is, I need an analogue of the statement (which only makes sense if such an $x_0$ exists):

There exists $A>0$ and $B<\infty$ such that $$A\leq \left|\frac{f(x)}{g(x)}\right|\leq B$$ for all sufficiently large $x$. Equivalently, $f(x)=O(g(x))$ and $g(x)=O(f(x))$ as $x\rightarrow\infty$.

By ``sensible'', I require that the analogous statement firstly implies the above whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$ and, secondly, that it does not imply the stronger statement $|f(x)/g(x)|\rightarrow C$ as $x\rightarrow \infty$ whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$.

The application I have in mind is number theoretic. Specifically, I am interested in the relative asymptotics of $L(x)$ and $M(x)$.

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.

I need a sensible definition of asymptotic equivalence when there is not necessarily an $x_0$ such that $f$ and $g$ are non-vanishing for all $x>x_0$.

That is, I need an analogue of the statement (which only makes sense if such an $x_0$ exists):

There exists $A>0$ and $B<\infty$ such that $$A\leq \left|\frac{f(x)}{g(x)}\right|\leq B$$ for all sufficiently large $x$. Equivalently, $f(x)=O(g(x))$ and $g(x)=O(f(x))$ as $x\rightarrow\infty$.

By ``sensible'', I require that the analogous statement firstly implies the above whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$ and, secondly, that it does not imply the stronger statement $|f(x)/g(x)|\rightarrow C$ as $x\rightarrow \infty$ whenever $f$ and $g$ are non-vanishing for all sufficiently large $x$.

The application I have in mind is number theoretic. Specifically, I am interested in the relative asymptotics of $L(x)$ and $M(x)$.