When we say a limit exists, does $\infty$ counts as existence? This is such a basic question but I'd like to get some clarify. When we say the "limit exists" for $\lim_{x \rightarrow c} f(x)$, does it count when $\lim_{x \rightarrow c} f(x) = \infty$?
 A: Usually, no, when we say a limit exists, that includes a requirement that it is finite. Such limits can, for instance, sensibly be used in further calculations, so it is a very natural distinction to make.
Compared to a limit of $\infty$, which is still, in some sense, a definite answer to what the limit is, but it's not really a sensible result as an intermediate calculation.
Ultimately, though, it is a matter of taste. And some do include infinite limits in the limits that they say exist. Any author ought to make it clear to their readers which way they mean it.
A: My opinion:
What we should say is like $\lim_{x \to 0} \frac 1 x$ does not exist, but $\lim_{x \to \infty} x = \infty$. But fine I guess I can put up with the convention in saying that $\lim_{x \to \infty} x$ does not exist because technically $\lim_{x \to \infty} x$ doesn't exist in the set of real numbers. It's also teaching students the idea like 'infinity isn't a number. it's a direction.' (Yeah right! 3 years later, you'll teach students infinity IS a number. See 'extended real number' below for more info.)
But...afaik, as $x \to$ any real number or $\pm \infty$, any function $f:\text{(insert appropriate domain)} \to \mathbb R$ will either approach a real number, approach $\infty$, approach $-\infty$ or not approach anything. There's nothing really more than $\mathbb R \cup \pm \infty$ that such a limit could approach, so I don't see the problem with counting $\pm \infty$ as existing. It's not like we could get a limit to be a (strictly) complex number or whatever.

My experience/answer:
In lower maths:

*

*Both $\lim_{x \to 0} \frac 1 x$ and $\lim_{x \to \infty} x$ count as 'does not exist' even if the reasons are different.

In (some) higher maths:


*$\lim_{x \to 0} \frac 1 x$ does not exist, but $\lim_{x \to \infty} x = \infty$.


*Eg In probability when we say like $\sum_{n} X_n = \infty \ a.s.$ Or in measure theory when we talk of conventions like  $0 \cdot \infty = 0$.


*Explanation: The thing is all higher maths begins with advanced calculus/elementary real analysis, where we define 'extended real number' to mean any real number or $\pm \infty$ (where $\pm \infty$ are defined) and even make precise the definition of $\lim_{x \to \text{anything}} f(x) = \infty$ (with the $\forall M > 0$).


*Detour to lower maths: Wait wait actually I think even some elementary calculus will talk about $\lim_{x \to \text{anything}} f(x) = \infty$! So yeah, all the more it really irritates me that $\pm \infty$ counts as merely 'does not exist' even in the lower maths. (My resolution is to say instead 'does not exist as a real number', but I guess elementary calculus students will find this weird like 'What else would it exist as?')


*Counter-eg (back to higher maths): Unfortunately, the opposite happens with the definition of the term 'integrable'. Here, just see real quick that the definition of 'Lebesgue integrable' is when some integral (don't mind the details) is '$< \infty$'. Basically 'Lebesgue integrable' means the '(Lebesgue) integral exists'. So limits existing is something like 'limitable'. Anyway, the meaning of '$< \infty$' is that the integral 1st exists as an extended real number and 2nd exists as a real number. Now, notice the weirdness here, the definition of 'Lebesgue integrable' is for (finite) real number, and yet they use the symbol '$\infty$'. WOW. Why don't they just say exists instead of '$< \infty$'? Seems pretty inconsistent. (My resolution is to call this thing 'finite Lebesgue integrable', but I guess people will find this annoying.)


*Wait going back to lower maths: Actually wait the same thing occurs with 'Riemann integrable'. In elementary calculus, an integral doesn't exist (recall an integral is a limit too!) if the 'limit' is $\infty$. Right now I can't quite think of a good example besides $\int_{1}^{\infty} 1 dx$. Maybe $\int_{1}^{\infty} \frac{1}{x} dx$. But anyway there are some examples of integrals that aren't Riemann integrable because the integral doesn't approach anything (in extended reals), similar to $\lim_{x \to 0} \frac{1}{x}$. Those are the ones that really 'don't exist'.
