Order Notation - Why can't $c$ be in terms of $n$?
Suppose we define $f(x) = O'(g(x))$ to mean that there is a function&nbsp;$c(x)$ such that $f(x)\leq c(x)g(x)$ for all sufficiently large&nbsp;$x$. Then $f(x) = O'(g(x))$ for almost every pair of functions $f$ and&nbsp;$g$, just by taking $c(x)=f(x)$. So, this definition just isn't a useful way to compare functions.

(I say "almost every pair" because there are issues if we have infinitely many values of&nbsp;$x$ for which $g(x)=0$ and $f(x)\neq 0$ but I don't think it's worth pinning down exactly what functions work and what don't.)