Infinite limits

Does a limit that has the value of infinite exist or not?

I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts what my calculus teachers and lecturers taught me however, that a limit doesn't exist if the right hand limit and left hand limit differ.

So which one is it?

Solution 1:

Obviously it depends on the definition of "exists". Some authors explicitly work over the extended real line with $\pm\infty$ adjoined, so that such infinite limits do explicitly "exist" as first-class values. But there is no consensus. One needs to pay attention to the author's definitions and conventions.

Perhaps it is worth mention - even though this case is rather trivial - that adjoining points at infinity is a special case of various constructions that attempt to simplify matters by some type of existential closure. Below I append an excerpt from my Oct 15, 1996 sci.math post.

This thread originated in a query as to whether infinity or $1/0$ could be admitted as a "value", and soon drifted into discussion of the Riemann sphere and other topological manifestations of infinity via compactification. Below I point out a couple of marvelous references on these topics; further I would like to bring to your attention a much wider perspective on such topics, namely that of existential closure as studied in model theory.

There is a beautiful exposition of points at infinity, projective closure, compactifications, modifications, etc. in [FM][1] Chapter 7, Points at Infinity, by H. Behnke and H. Grauert. This is volume III in the excellent "Fundamentals of Mathematics" series, which deserves to be on the bookshelf of every budding mathematician.

A much deeper appreciation of the methodology behind these constructions can be had by studying them from a model-theoretic perspective, in particular from the standpoint of existential closure and model completion. Kenneth Manders has written a series of thought provoking papers [2],[3] from this perspective. I close with an excerpt from the introduction to [2]:

"The systematic adjunction of roots, or solutions to other simple conditions, as in formation of the complex numbers by adjoining imaginaries, or in adjunction of points "at infinity" in traditional geometry, may be analysed as existential closure and model completion. 'Existential closure' refers to a class of processes which attempt to round off a domain and simplify its theory by adjoining elements -- more properly, it refers to the formal relationship that obtains in such a process. 'Model completion' is one of the terms employed when this process is successful. The formation of the complex numbers, and the move from affine to projective geometry, are successes of this kind. Thus, the theory of existential closure gives a theoretical basis of Hilbert's "method of ideal elements." I now sketch the theory of existential closure, to bring out when, how, and in what sense existential closure gives conceptual simplification."

[FM] Fundamentals of mathematics. Vol. III. Analysis.
Edited by H. Behnke, F. Bachmann, K. Fladt and W. Suss.
Translated from the second German edition by S. H. Gould.
Reprint of the 1974 edition. MIT Press,
Cambridge, Mass.-London, 1983. xiii+541 pp. ISBN: 0-262-52095-8 00A05

[2] Manders, Kenneth
Domain extension and the philosophy of mathematics.
J. Philos. 86 (1989), no. 10, 553--562.
http://www.jstor.org/stable/2026666

[3] Manders, Kenneth L.
Logic and conceptual relationships in mathematics.
Logic colloquium '85 (Orsay, 1985), 193--211,
Stud. Logic Found. Math., 122,
North-Holland, Amsterdam-New York, 1987.
http://dx.doi.org/10.1016/S0049-237X(09)70554-3

Solution 2:

It is important to consider the context in which you are taking the limit. When taking the limit of a sequence $\{ x_{n}\}$, we must consider what set the elements of that sequence are coming from. For example, if $\{ x_{n}\} \subset \mathbb{R}$, then for the limit $L$ to exist (in $\mathbb{R}$) we must have that $L \in \mathbb{R}$. If $\{ x_{n}\} \to \infty$ as $n \to \infty$, then the limit doesn't exist (in $\mathbb{R}$).

But, there is another notion of limit called $\text{lim sup}$. According to Baby Rudin (slightly modified):

3.16 Definition Let $\{s_{n}\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ (in the extended real number system [which includes $\pm \infty$]) such that $s_{n_{k}} \to x$ for some subsequence $\{s_{n_k}\}$. This set $E$ contains all subsequential limits* plus possibly the numbers $+ \infty$, $-\infty$.

Now define $$\begin{align}\text{lim sup} \;\; s_{n} &= \sup E, \;\;\;\;\text{and} \\\ \text{lim inf} \;\; s_{n} &= \inf E.\end{align}$$ * A subsequential limit is just the limit of the subsequence, if that subsequence converges.

With this definition we can discuss infinite limits (superior or inferior).

EDIT: Removed my erroneous claim that finite $\text{lim sup}$'s and $\text{lim inf}$'s collapse to the regular old $\lim$. (Counterexample: $\text{lim sup}_{x \to \infty} \sin(x) = 1$ but $\lim _{x \to \infty}\sin(x) \neq 1$.

Solution 3:

There is no ambiguity about infinite limits. For instance, when we define that a sequence or real numbers $(x_n)$ "tends to infinity", $(x_n) \longrightarrow +\infty$, we just mean that, for each real number $N$ there is some $n_0 \in \mathbb{N}$ such that, for all $n\geq n_0$, we have $x_n > N$. That is, the sequence $(x_n)$ grows indifinitely. There is no implication here that $+\infty$ "exists", or doesn't, nor any problem with the fact that, certainly, $+\infty$ is not a real number. In fact, there is no $\infty$ on the right hand side of the definition, but just real numbers. And symbols on the left hand side of a definition mean just you want them to mean according with what you put on the right hand side.