Why is it impossible to find a nontrivial example of someone using little-oh notation?
Recall, little oh is defined as follows. We write $ f(x) = o(g(x))$ if
$ \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$.
Let’s say I want to get rid of silly little terms like $ \log \log x$. Notice that:
$ \log\log x = e^{\log\log\log x} = e^{(\log\log x) \frac{\log \log \log x}{\log \log x}} = (\log x)^{\frac{\log \log \log x}{\log \log x}}$.
Of course, $ \lim_{x \to \infty} \frac{\log \log \log x}{\log \log x} = 0$, so this exponent is $ o(1)$.
Using this trick, we can rewrite expressions in this way:
$ \log x \log \log x = (\log x)^{1+o(1)},$
and hence all the ugly little log terms have been swept under the rug.