4. Multiple variables
The asymptotic notations can be extended to the case where the functions depend on multiple variables. E. g. if we have an algorithm for searching a string of length $m$ in a text of length $n$, we might be interested in how the worst-case running time $T(m,n)$ behaves when both the search string and the text get very long (i. e., when $m$ and $n$ both grow to infinity).
The definitions of asymptotic notations are extended to the multivariate case in a natural way. E. g. the definition of Big Oh for two variables going to infinity sounds like this:
For example, if $T(m, n) = 4mn$, $g(m, n) = mn^2$ and $h(m, n)=n^3$, then $T \in O(g)$ but $T \not\in O(h)$.
Whenever multiple variables are involved, one must to be careful to make clear which variables are function arguments that go to infinity, and which are constants: for instance, the statement $nk\in O(n)$ is ambiguous, if we don't know whether $k\to \infty$ or $n\to\infty$ or both. If $n \to \infty$ and $k$ is a constant, then the statement $nk\in O(n)$ is true; if $n$ and $k$ both go infinity or only $k$ goes to infinity, then it is false.