<s>[INST] Trying to find $c_{1}$ , $c_{2}$ for Big theta notation. Need to find $c_{1},$ $c_{2}$ and $n_{0}$.
\begin{equation}
c_{1}n^3 \leq \frac{n^3}{100} - 100n^2 - 100n + 3\leq c_{2}n^3
\end{equation}
\begin{equation}
c_{1} \leq \frac{1}{100} - \frac{100}{n} - \frac{100}{n^2} + \frac{3}{n^3} \leq c_{2}
\end{equation}
\begin{equation}
\frac{1}{100} - \frac{100}{n} - \frac{100}{n^2} + \frac{3}{n^3} \leq c_{2}
\end{equation}
This is what I have done so far, but I am currently stuck do I need to chose c2 which is positive and greater than $\frac{100}{n} + \frac{3}{n^3}$? And for c1 , I need a c1, which is >0. How do I find these constants? 
 [/INST] As in polynomials when the variable goes to $\infty$ their behavior become like their greatest power term, so the inequality use this fact. But as $n$ becomes greater $c_1,c_2$ become more closer to $\frac{1}{100}$. Therefore if you know the bound of $n$, then you can far $c_1,c_2$ away from $\frac{1}{100}$ as it allows you.
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