# Thread: Θ-classes to arrange functions?

1. ## Θ-classes to arrange functions?

Use the rules and definition for ordering Θ-classes to arrange the following in order from lowest to highest:
lg n ; (lg n)^6 ; lg(n^n) ; n^4 ; n^(1/2) ; 5n^3+n^2 ; 1^n ; (1.001)^n ; 5^n

I used the rules as I understood them and ended very uncertainly with the following order:

1^n (1 to the power of anything is)
lg n
n^(1/2)
lg (n^n)
5n^3+n^2
n^4
(lg n)^6
(1.001)^n
5^n

I'm really not sure about the order so if someone could please check and tell me where and why I went wrong I would really appreciate it.

2. The general rule is that for every $a_1,a_2>1$ and $\beta_1,\beta_2,\beta_3>0$, for the functions $f_1(n)=(\log_{a_1} n)^{\beta_1}$, $f_2(n)=n^{\beta_2}$ and $f_3(n)={a_2}^{{\beta_3} n}$, we have $\lim_{n\to\infty}f_1(n)/f_2(n)=0$ and $\lim_{n\to\infty}f_2(n)/f_3(n)=0$. So you answer is correct except for the place of $(\lg n)^6$.
By the way, did you realize that for $n=1000\,000$, $n^4=10^{24}$ (the order of the Earth's mass in kilograms), while $(\lg n)^6<50\,000$ (decimal logarithm)?