I want to proof for a given function, that some other function is an upper bound of the given function. For example, given is the function:
$f (n) = \left\{ \begin{array}{lll} \frac{1}{2log(n)}-2n+2^n &\textrm{n>42 and odd} \\ (\frac{7}{2})^n + \frac{3}{4}n^2 - log(n) & \textrm{n ≤ 42} \\ \frac{3^n}{n^3} - (n+1)^3 + log(2^n) & \textrm{otherwise}\\ \end{array} \right.$
I now want to check with the Big O definition:
$\lim\limits_{n \rightarrow \infty}{\frac{f(n)}{g(n)} < \infty}$
if $n^2$ or $e^n$ is an upper bound or not.
This should be an easy task, however I'm also a little bit unsure how to handle the cases. If I can proof that one case is not an upper bound, does this proof that the whole function can't be an upper bound?
In the three cases you present, there are actually only 2 that interest us, the first and third.
This is because you want to test for asymptotic limits when $n$ becomes large, and the second case only deals with the first elements of the function, so it's not important for the limit.
However, for the first and last case, then yes, both of them have to be bounded in order to be able to say that $f=O(g)$.
The way I simplify it in my mind is that I just think of it as a sequence $u_n$ defined via $f$. Then you do have to bound all the cases to bound the sequence.