Properties of Asymptotic analysis

Question

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 $f_1(f_2)\sim g_1(g_2)$?

Answer 1

It is compatible only with

• limits: if $\ell\ne0$, $\displaystyle\lim_af=\ell\iff f\sim_a=\ell$
• products
• integer powers
• inverse
• quotients
• antiderivative: more precisely, if $f\sim_ag$, then $\displaystyle\int_a^xf(t)\,\mathrm d\mkern1mut\sim_a\int_a^xg(t)\,\mathrm d\mkern1mut$
• right composition by the same function: if $\;\displaystyle\lim_a g=b$ and $f_1\sim_bf_2$, then $\;f_1\circ g\sim_a f_2\circ g$
• left composition by logarithm: if in a neighbourhood of $a$, $\dfrac1{\ln\circ g}\;$ is bounded, and $f\sim_ag$, then $\;\ln\circ f\sim_a\ln\circ g$.
• left composition by the exponential function: if $\;\displaystyle\lim_a(f-g)=0$, then $\;\mathrm e^f\sim_a\mathrm e^g$.

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 $f'(x)=2\nsim_0g'(x)=1$.

Answer 2

No, it doesn't. For instance, $f_1(x) = x^2$ and $f_2(x) = x^2 - x$; $g_1(x) = g_2(x) = x^2$. Then look at $$ (f_1 - f_2) (x) = x \\ (g_1 - g_2) (x) = 0 $$ which are clearly not asymptotically equivalent.