Asymptotic behavior of $n\sqrt n + n \log n$ & $\log_{100} n$ [duplicate]

I have the following two functions

• $$f(n) = n\sqrt n + n \log n$$
• $$\log_{100} n$$

And I need to classify them into the followings:

• $$O(n)$$, and/or
• $$O(n^2)$$, and/or
• $$O(n^3)$$, and/or
• $$O(n^{1.5})$$, and/or
• $$\omega(n)$$, and/or
• $$o(n\log n)$$ (small o)
• $$\theta(n^{1.5})$$,

Regarding the $$\theta(n^{1.5})$$, I am grasping it's idea newly, where do these functions set in. Also, does the function $$f(n) =n^{0.9999}$$ belong to $$\theta(n^{1.5})$$.

I would appreciate your feedback, as from your answers I am building up the knowledge for these.

• Am I supposed to define w(n) by myself? – John L. Oct 17 '18 at 18:22
• it is possibly $\omega(n)$ – kelalaka Oct 17 '18 at 18:32
• Welcome back to Computer Science! You can use LaTeX to typset mathematics, rather than doing it manually with sup tag and some other weird stuff. I edited to show you how; we also have a brief tutorial. – John L. Oct 17 '18 at 18:39
• @kelalaka, thanks for the hint. tony9099, please edit if I have not updated your question correctly. – John L. Oct 17 '18 at 18:41

• $$\log_{100}n \in \mathcal{O}(log(n))$$ since base change is a constant.
• $$\log n \in \mathcal{O}(n)$$
• $$\log n \in \mathcal{O}(n^2)$$
• $$\log n \in \mathcal{O}(n^3)$$
• $$\log n \in \mathcal{O}(n^{3/2})$$
• $$\log n \not\in \mathcal{\omega}(n)$$
• $$\log n \in \mathcal{o}(n \log n)$$

This table gives the order;

First of all;

$$\lim_{n \rightarrow \infty} \frac{n \sqrt n}{n \log n} = \infty$$ by using this;

• $$f(n)= n\sqrt n + n\log n \in \mathcal{O}(n\sqrt n )$$.
• $$n\sqrt n \not\in \mathcal{O}(n)$$
• $$n\sqrt n \in \mathcal{O}(n^2)$$
• $$n\sqrt n \in \mathcal{O}(n^3)$$
• $$n\sqrt n \in \mathcal{O}(n^{3/2})$$
• $$n\sqrt n \in \mathcal{\omega}(n)$$
• $$n\sqrt n \not\in \mathcal{o}(n \log n)$$,
• Please remember that we're not a homework answering service. Answers that don't teach people how to solve the problem aren't very valuable at all. – David Richerby Oct 17 '18 at 19:51
• Most questions of the for "Is this function big-O of this other function" can be flagged as duplicates of the one I marked. That question covers the general techniques and there's usually not much more to say, since the same functions tend to be used as examples each time. – David Richerby Oct 17 '18 at 20:13
• @kelalaka Thanks a lot for your input :) I have edited the question and added a new bound as well as a new function. I would appreciate if you can have a look :) (PS: this is not a homework) – tony9099 Oct 18 '18 at 16:36
• Take the limit at WolframAlpha. That easy. – kelalaka Oct 18 '18 at 16:39