Big O Notation / landau symbols

Question

I want to write these two in the big O notation: (it's $h\rightarrow0$)

1. $f(h)=\sqrt{h^3}$

2. $f(h)=h\cdot \log h $

But I don't have any idea how to do this.

Thanks for helping!

Answer 1

What you ask makes no sense, really. You have two functions; they are fine as they stand, no big O is needed to understand them. Big O is a notation to related a given function to something else, presumably simpler, or just to write an estimate in a form where you don't want to bother with the implied constants. You could certainly write $\sqrt{h^3}=O(h)$, for example, but if it makes sense to do so depends on context. Ditto for the other one, which is (among many things) $h\cdot\log(h)=O(h^{1+\varepsilon})$ for any $\varepsilon>0$. But again, whether it makes sense to write it so depends on context.