I have a simple question.
We usually use ~ notation if two functions are asymptotically equivalent.
My question is, does summation, subtraction, multiplication, and composition of functions preserve the ~ notation?
i.e.
if $f_1(x)\sim g_1(x)$ and $f_2(x) \sim g_2(x)$, is it true that $f_1+f_2 \sim g_1+g_2$,
$f_1-f_2 \sim g_1-g_2 $,$f_1*f_2 \sim g_1*g_2$, and $f_1(f_2)\sim g_1(g_2)$?
Best Answer
It is compatible only with
It is not compatible with sums: $x+x^2\sim_0 x$, $-x+x^3\sim_0-x$, but $(x+x^2)+(-x+x^3)\nsim_0 0$.
It is not compatible with derivatives either: $f(x)=1+2x\sim_01\sim_0g(x)=1+x$, but $f'(x)=2\nsim_0g'(x)=1$.