Talk:Limit of Sequence is Limit of Real Function

I'm fairly sure this one's already up. Can't immediately find it, I'll need to look for it. --prime mover 18:50, 27 January 2012 (EST)

I wasn't able to find it, could be. LF, why do you think it's definition rather than a theorem? It's not at all obvious to me that you can change the domain of a function and still guarantee $\forall \epsilon \exists M$. --GFauxPas 19:01, 28 January 2012 (EST)

Well, I anticipated problems with functions like:

$f(x) = \begin{cases}x &: x \notin \N\\0&: \text{otherwise}\end{cases}$

for which I can't see a sensible definition of $\displaystyle\lim_{n\to\infty} f(n)$ other than to interpret it as a sequence.

It could also be that you meant to write $\displaystyle\lim_{x\to\infty} f(x)$ instead (which I think, upon reading again), in which case I haven't said anything. However, it is advisable to distinguish variables from different domains, as it inevitably leads to confusion (as you have seen). --Lord_Farin 08:40, 29 January 2012 (EST)

That's why I said "let f be a real function", to emphasize that the domain could be $\R$. This theorem is quite a useful one though when you want to use L'Hopitals Rule to evaluate the limit of a sequence. Without this theorem, using LHR wouldnt be justified because a sequence isn't differentiable. --GFauxPas 09:00, 29 January 2012 (EST)

I think I confused you because I used n as the argument. Do people associate n that strongly with natural numbers? It's just that I liked the symmetry of $f(n) = a_n$, but $f(x) = a_n$ is the same thing, or perhaps $f(x) = a_x$. --GFauxPas 09:05, 29 January 2012 (EST)

Well, you can say $n \in \R$, you can say $n \in \N$, but you can't expect people to get that you mean both at the same time. If $n \in \N$ somewhere on the page, it is very likely that people will assume such is the case everywhere. Same with $x$ instead of $n$. So, what I propose is, that you say $\lim_{x\to\infty} f(x)$ to emphasize the real nature of the limit, and also write $x\mapsto f(x)$. That would remove any confusion I think. --Lord_Farin 09:14, 29 January 2012 (EST)