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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.

Prem
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turkey131
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    The problem is limits don't commute in general. This question uses two $\to0$ limit operators rather than $m$ limits that $\to\infty$, but the principle is the same. So you need to clarify, e.g. with a statement that you're numbering the variables so $\lim_{\substack{x_i\to\infty}\i=1,\cdots,m}$ means $\lim_{x_1\to\infty}\lim_{x_2\to\infty}\cdots\lim_{x_m\to\infty}$. – J.G. Mar 24 '24 at 15:14
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    ((1)) You can not use it in general because the limit might depend on the order for the general class of functions ((2)) When you are considering a special class of functions where the order is immaterial , you can & should use the compact way to high-light that the order is immaterial. – Prem Mar 24 '24 at 16:08

2 Answers2

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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 $$ \lim_{x_1\to\infty}\lim_{x_2\to\infty}f(x_1,x_2)=\infty\neq1=\lim_{x_2\to\infty}\lim_{x_1\to\infty}f(x_1,x_2). $$

stochastic-conch
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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 $$\lim_{x_2\to\infty} \lim_{x_1\to\infty} f(x_1,x_2)$$

jjagmath
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