Does $\lim_{q\to \infty }f(q)$ has sense where $q\in \mathbb Q$?

Question

I wrote in a previous exam $$\lim_{\underset{q\in \mathbb Q}{q\to \infty} }f(q)=1\neq 0 =\lim_{\underset{r\in \mathbb R\backslash \mathbb Q}{r\to \infty }}f(r),$$ but my teacher told me that such limit has no sense, but I don't understand why. Is that really have no sense ? And if yes, why ?

Answer 1

Suppose $E\subseteq A\subseteq\Bbb R,$ $f:A\to\Bbb R,$ and $a,L\in\Bbb R.$

When we say $$\lim_{\underset{x\in E}{x\to \infty} }f(x)=L,$$ we mean that $E$ (and so $A$) is unbounded above in $\Bbb R,$ and that $$\forall\epsilon>0,\exists M:\forall x\in E,x>M\implies\bigl|f(x)-L\bigr|<\epsilon.$$

When we say $$\lim_{\underset{x\in E}{x\to a} }f(x)=L,$$ we mean that $a$ is a limit point of $E$ (and so of $A$), and that $$\forall\epsilon>0,\exists \delta>0:\forall x\in E,0<|x-a|<\delta\implies\bigl|f(x)-L\bigr|<\epsilon.$$

Both are perfectly sensible. We can also talk about one-sided $E$-limits, and limits as $x$ decreases without bound in $E$.

Now, it's possible that limits of this sort weren't part of the context of your course, so "don't make sense" as far as the given definitions of the course are concerned.

It could also be that the definitions are in the context of the course, but that your answer doesn't really address the question--without knowing what the question was, that's impossible for anyone here to say.