Could you help me to show: $g(x,\epsilon)+f(x,\epsilon)=O(|\phi(x,\epsilon)|+|\psi(x,\epsilon)|)$ but $g(x,\epsilon)+f(x,\epsilon)\neq O(\phi(x,\epsilon)+\psi(x,\epsilon))$ (both when $\epsilon\to0$), where $O$ stands for the big Oh notation?
How to prove asymptotic property with big O notation?
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1Try $f(x)=g(x)=\phi(x)=x$ and $\psi(x)=-x$. – 2011-02-21
1 Answers
So, assuming $g(x,\epsilon)=O(\phi(x,\epsilon))$ and $f(x,\epsilon)=O(\psi(x,\epsilon))$ this means that $\exists \delta_1,\delta_2,M_1,M_2$ such that
$\forall |\epsilon|<\delta_1 : |g(x,\epsilon)| < M_1 |\phi(x,\epsilon)|$
and
$\forall |\epsilon|<\delta_2 : |f(x,\epsilon)| < M_2 |\psi(x,\epsilon)| \; .$
This implies that
$\begin{eqnarray} \forall |\epsilon| < \min(\delta_1,\delta_2) : & |g(x,\epsilon) + f(x,\epsilon)| & \leq |g(x,\epsilon)| + |f(x,\epsilon)| \\ & & < \max(M_1,M_2)(|\phi(x,\epsilon)| + |\psi(x,\epsilon)|) \; , \end{eqnarray} $
using the triangle inequality. Or in other words $g(x,\epsilon)+f(x,\epsilon)=O(|\phi(x,\epsilon)|+|\psi(x,\epsilon)|)$.
Now, using Yuval Filmus' suggestion, you can see that for the choice $f(x)=g(x)=\phi(x)=x$ and $\psi(x)=-x$, $2x=O(2|x|)$, which is correct. The other identity can not be correct however since $2x \neq O(0)$.