Limit notation (for taking multiple limits at once)

Question

This is probably a silly question, but I am interested in looking at limits of multi-variable functions, such as \begin{equation}\lim_{x_1\to\infty}\lim_{x_2\to\infty}\cdots\lim_{x_m\to\infty}f(x_1,\cdots,x_m).\end{equation}

Is it notationally acceptable to simply write the above as \begin{equation}\lim_{\substack{x_i\to\infty}\\i=1,\cdots,m}f(x_1,\cdots,x_m)?\end{equation}

This notation would be slightly less space-consuming which is why I thought it would be a good idea but I'm not sure if this is standard notation.

Answer 1

The notation you suggest might be ambiguous, since the limits might not commute, i.e. we might have (in the case $m=2$ here) $$ \lim_{x_1\to\infty}\lim_{x_2\to\infty}f(x_1,x_2)\neq\lim_{x_2\to\infty}\lim_{x_1\to\infty}f(x_1,x_2). $$ To give an example, consider $$ f(x_1,x_2)=\bigg(1+\frac{1}{x_1}\bigg)^{x_2}, $$ with $x_1,x_2>0$. Then $$ \lim_{x_1\to\infty}\lim_{x_2\to\infty}f(x_1,x_2)=\infty\neq1=\lim_{x_2\to\infty}\lim_{x_1\to\infty}f(x_1,x_2). $$

Answer 2

This notation could lead to confusions. Some may think that $$\lim_{\substack{x_i\to\infty\\i=1,2}}f(x_1,x_2)$$ means $$\lim_{x_1\to\infty} \lim_{x_2\to\infty} f(x_1,x_2)$$ while others could think it means $$\lim_{x_2\to\infty} \lim_{x_1\to\infty} f(x_1,x_2)$$