I have trouble starting this homework assignment for my discrete math class. Does $\Theta(n^3+2n+1) = \Theta(n^3)$ hold? Justify your answer.
We need to prove $n^3+2n+1 \in \Theta(n^3)$ AND $n^3 \in \Theta(n^3+2n+1)$\
First lets solve $n^3+2n+1 \in \Theta(n^3)$
This is where I'm getting stuck. I'm supposed to do something with transitivity but I don't entirely know how to do that. Do I use the definitions like:
$l|n^3| \leq n^3+2n+1 \leq k|n^3|$ let $l = 1$ and $k=1$
and so on?
Any tips would be helpful.
Note that by definition $$ f\in O(g)\text{ as }x\rightarrow a $$ whenever $$ \limsup_{x\rightarrow a}\left|\frac{f(x)}{g(x)}\right|<\infty $$ in this case since
$$\dfrac{n^3+2n+1}{n^3}\to1$$
we have that $n^3+2n+1∈O(n^3)$ and $n^3∈O(n^3+2n+1)$.