Suppose we define big$-$O complexity as:

$O(g(n))=$ set of all functions $f(n)$ such that

$\underset{n}{lim}\frac{f(n)}{g(n)}=c$

where $c0$ is a constant. Will it work? List $3$ good examples of pairs of functions where

this will work. Also, show a counter example $($pair of functions$)$ where this definition does

not agree with the one given in class.