I was just dealing with a problem - actually it was a question on stats.SE - of some asymptotic results and while doing so I asked myself whether we are free of doing some algebraic manipulations under which the equivalence relation is invariant:
Let's science up: say we have two functions $f$ and $g$ and we define an equivalence relation $$ f\sim g \iff \lim_{n\to \infty}\frac{f(n)}{g(n)}=1 $$
could we now also say for example $$ f+1\sim g+1 \text{ or } \log f\sim \log g \tag1 $$ the specific question I had in mind was whether $$ 1-f\sim g \iff f\sim 1-g \tag{2a} $$ holds or not. Which means in the actual problem setting of the above linked question, if we can deduce any information from $$ 1 - \Phi(x) \sim\frac{\phi(x)}{x} \tag{2b} $$ with $\Phi(x)$ CDF of the standard normal and $\phi(x)$ corresponding density, to $$ \Phi(x)\sim \;? \tag3 $$Sure there are a lot of examples where $(1)$ holds, but I am not sure if this is generally true. Also $(2)$ is definitely not true in general (take $x-1$ and $x$) - is it maybe true in this special case? Can we maybe slightly modify the above relation to make it being invariant under such operations?