Compact notation for limit at two different points of interest

Question

Is it acceptable to replace $$ \lim_{x\to a}f(x)=\lim_{x\to b}f(x)=c $$ with $$ \lim_{x\to \{a,b\}}f(x)=c? $$ If not, is there another way I could state such a fact in a compact manner without having to write out the limit twice?

Edit: Another viable option I considered was $$ \lim_{x\to a\lor b}f(x)=c. $$ I am hoping there is some notation for this that won't require any explanation prior to using it in a proof.

Answer 1

Both variants are not commonly used notations and have drawbacks.

• When looking at \begin{align*} \lim_{\color{blue}{x\to \{a,b\}}}f(x)=c \end{align*} we consider the limit of a point $x$ to a set $\{a,b\}$. This is not the intention here and it's not obvious, that elements of the set should be taken instead.

• The other variant \begin{align*} \lim_{\color{blue}{x\to a\lor b}}f(x)=c \end{align*} uses a boolean operator $\lor$, but $a$ and $b$ are not logical expressions.

I think a useful proposal to reduce the number of limits is stated in the comment section by @JWTanner. \begin{align*} \lim_{\color{blue}{x\to L}}f(x)=c\qquad\qquad \color{blue}{L\in\{a,b\}} \end{align*} This is mathematically precisely what we want to say and we also keep common notation.