Definition
35
[1FB]Let \(a∈\overlineℝ\) and \(I\) be a neighborhood of \(a\). Let \(f,g:I→ℝ\). We will say that ”\(f(x)=o(g(x))\) for \(x\) tending to \(a\)” if 1
\[ \forall \varepsilon {\gt}0,~ \exists \delta {\gt}0 ,x\in I \land ~ |x-a|{\lt}\delta \Rightarrow |f(x)|\le \varepsilon |g(x)| \quad . \]
This notation reads like ”f is small o of g”.
If \(g(x)≠ 0\) for \(x≠ a\), then equivalently we can write
\[ \lim _{x→ a}\frac{f(x)}{g(x)}=0\quad . \]
We will say that ”\(f(x)=O(g(x))\) for \(x\) tending to \(a\)” if if there is a constant \(c{\gt}0\) and a neighborhood \(J\) of \(a\) for which \(∀ x∈ J, |f(x)|≤ c |g(x)|\).
Again, if \(g(x)≠ 0\) for \(x≠ a\), then equivalently we can write
\[ \limsup _{x→ a}\frac{|f(x)|}{|g(x)|}{\lt}∞\quad , \]
This notation reads like ”f is big O of g”.
For further information, and more notations, see [ 45 ] .