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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!

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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.

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I don't really why $h\rightarrow 0$, i use this notation when $h\rightarrow\infty$ for look the complexity of algorithm. So, may be you want to use a substitution with $u=1/h$ and see it like me.

  1. $O(h^{1/3})$

  2. $O(n\cdot \log n)$ look that is Big O, is like that Complexity for Quicksort Algorithm in the average case.

Big-O is for me like a function boundary for the function and take the most important for the result function. Then, we forget things like coefficient constants. For Example: $O(4x^4+ x^2)=O(4x^4)= O(x^4)$

Regards,

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