I'm proving the statement of some limit which has a form of
$$\lim_{\|\mathbf{m}\| \to \infty} f(m_{1},m_{2},\cdots,m_{k}) = S$$
where $\mathbf{m} = (m_{1}, \cdots, m_{k}) \in \mathbb{R}^{k}$.
I've proved that statement for case that $m_{i} \to \infty$ for each $i$, and being working on the remaining cases.
My question is, HOW CAN I CHARACTERIZE the remaining cases rigorously?
"The remaining cases" is a bold understatement. For example, if $f(m_1,\ldots,m_n)=S+\frac{1}{1+\sum_{1\le i<j\le n}(m_i-m_j)^2}$ then the limit is $S$ if $m_i\to\infty$ and the other components remain constant, but it is $S+1$ along the line $m_1=\ldots =m_n$. Exceptional cases like this, but of arbitrary complexion, can easily be constructed. So, ultimately you have no choice but to hava look at the specific properties of $f$ and show for each $\epsilon>0$ that $|f(\mathbf m)-S|<\epsilon$ if $\|\mathbf m\|>M$ for some $M$.