I'm struggling with how to choose witnesses for asymptotic growth. Specifically, here is the problem I'm working on and what I have done so far:
Prove: $$4n^5 – 50n^2 + 10n \in \Theta(n^5)$$
$$0 \leq c_1(n^5) \leq 4n^5 – 50n^2 + 10 n \leq c_2(n^5)$$ $$0 \leq c_1 \leq 4 – 50/n^3 + 10/n^4 \leq c_2$$
Now, I know for the middle portion, so long as $n \geq 3$ then it will approach 4, leaving me with
$$0 \leq c_1 \leq 4 \leq c_2$$
However, I don't understand how to get my exact witnesses. Everything I found on the internet, people choose their witnesses but don't really explain how. Can I just arbitrarily choose anything between 0 and 4 for $c_1$ and anything above 4 for $c_2$?
A polynomial will always asymptotically behave like the highest degree monome. This can easily be proven:
Take for $k\leq n$ $$ \lim_{x\to\infty} \frac{a_kx^k}{x^n} = \lim_{x\to\infty} a_k x^{k-n} $$ If $k=n$ this is simply $a_k$, if $k<n$ this goes to $0$. Thus $$ \frac{a_0+a_1x+\ldots+a_nx^n}{x^n} \xrightarrow{x\to\infty} a_n $$