Definition 1.1 (Big $O$ and little $o$ notation). Let $f,g,h:S\to ℂ$, $S\subseteq ℂ$.

Write $f(x)=O(g(x))$ if there is $c>0$ such that $|f(x)|\le c|g(x)|$ for all $x\in S$.

Write $f(x)=o(g(x))$ if for any $𝜀>0$ there is $x_{𝜀}>0$ such that $|f(x)|\le 𝜀|g(x)|$ for $x\in S$, $|x|\ge x_{𝜀}$.

Write $f(x)=g(x)+O(h(x))$ if $f(x)-g(x)=O(h(x))$ and write $f(x)=g(x)+o(h(x))$ if $f(x)-g(x)=o(h(x))$.