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Consider the following limits:

$$ \lim_{x\rightarrow0}\frac{1}{x^2}$$

$$ \lim_{x\rightarrow0}\frac{1}{x}$$

As far as I can tell, most authors say as a matter of terminology that these limits don't exist and are not defined, since they don't have real-number values. For the first one, we can write

$$ \lim_{x\rightarrow0}\frac{1}{x^2}=\infty,$$

whereas I think most authors would say that it was false that

$$ \lim_{x\rightarrow0}\frac{1}{x}=\infty,\qquad \text{(FALSE)}$$

since the one-sided limits have opposite signs. What is the most standard way to verbally describe the second situation in contradistinction to the first? It's true that the limit doesn't exist and is not defined, but how do you communicate, without being too cumbersome, the fact that it's even more naughty and bad than the first one? I'm hoping there's some verbalism that's not as clumsy as "a limit that is infinite and has the same sign from both sides" versus "a limit that is infinite and has opposite signs from the two sides."

Several comments and answers have claimed that $\lim 1/x^2$ is described verbally as being defined or existing, while $\lim 1/x$ is described as being undefined or nonexistent. As far as I can tell from the paper and online sources I have handy, this is not standard. Standard terminology seems to be that both of these limits are considered nonexistent and undefined.

  • I often see "DNE" for "does not exist" – Ben Grossmann Jun 23 '14 at 23:51
  • I usually say the limit of the second type does not exist, whereas I allow a limit to exist and possibly be $+/- \infty$, specifing convergence if the limit exists as a real number. – Forever Mozart Jun 23 '14 at 23:51
  • @Omnomnomnom: As far as I can tell by looking at various sources, people describe both of these limits verbally as not existing. I'm asking for a verbalism that distinguishes the two cases. –  Jun 23 '14 at 23:53
  • @TomCruise: Are you saying you'd describe the second limit as existing but not converging? AFAICT from looking at various sources, that would be nonstandard. –  Jun 23 '14 at 23:54
  • I do the same as @TomCruise, so sometimes I'd say something like 'the sequence is increasing, so it has a limit' or 'this sequence has a finite limit' or 'this sequence has a limit (possibly infinite)'. this works pretty well for me and for a bunch of people I know. as to Ben's comment - I'd say the second limit doesn't exits, dunno about Tom tho' – mm-aops Jun 23 '14 at 23:56
  • I should have specified: it is sometimes the case that it would be considered wrong to say that the first limit does not exist – Ben Grossmann Jun 23 '14 at 23:57
  • @mm-aops: So how would you distinguish the two cases in the question? #1 "has an infinite limit," while #2 "has no finite or infinite limit?" –  Jun 23 '14 at 23:58
  • I'd say the second "has no limit" while I'd say that the first one "has a limit" I might then specify (or not) if it's finite. when I say "this sequence has a limit" I don't mean to say anything about its finiteness (is this even a word?). it's just that I sometimes underline that it might be infinite just to focus the reader's/listener's attention on it – mm-aops Jun 24 '14 at 00:01
  • @Omnomnomnom: I should have specified: it is sometimes the case that it would be considered wrong to say that the first limit does not exist OK, if the terminology is not totally standardized then that would be good to know. However, in the small sample of books I have handy on paper or online, all seem to define the first limit as not existing. –  Jun 24 '14 at 00:02
  • @BenCrowell I'll give you a source if I find an old calculus book lying around. Another useful piece of terminology that may apply here is saying that the first limit "diverges to infinity", as opposed to "simply" not existing – Ben Grossmann Jun 24 '14 at 00:10
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    @Omnomnomnom It probably would be better practice to say "diverges to (positive/negative) infinity", rather than using the somewhat perverse "infinite limit". But it seems to be common to avoid using terms like "convergence" and "divergence" in first-semester (or so) calculus. – colormegone Jun 24 '14 at 00:36
  • The left and right hand limits about zero are not equal...? – ClassicStyle Jun 24 '14 at 00:43
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    Here's an old, old thread on this topic: http://math.stackexchange.com/questions/3203/infinite-limits – colormegone Jun 24 '14 at 00:47

2 Answers2

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I would simply say that the function has opposite-signed infinite unilateral limits at the point, or the function has infinite unilateral limits of opposite signs at the point, where "unilateral" can be replaced with "one-sided" if you wish. Note that in such a case the absolute value of the function has an infinite bilateral (i.e. two-sided) limit at the point. However, it is possible for the absolute value of a function to have an infinite bilateral limit at a point without having either unilateral limit exist (finitely or infinitely), as for example is the case with the function $$ f(x)=\begin{cases} q & \text{if} & x = \frac{p}{q} \; \text{and} \;\; q \;\; \text{is even} \\ -q & \text{if} & x = \frac{p}{q} \; \text{and} \;\; q \;\; \text{is odd} \\ \frac{1}{x} & \text{if} & x \notin \mathbb Q \\ 0 & \text{if} & x=0 \end{cases} $$ at $x=0,$ where $p$ and $q$ are relatively prime nonzero integers and $q > 0.$

Dave L. Renfro
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We use $+$ and $-$ as subscripts. As in $$\lim_{x \to 0^+} \frac{1}{x} = + \infty \qquad \lim_{x \to 0^-} \frac{1}{x} = - \infty$$ A shorthand for non-existence is the symbol $\nexists$, so $\nexists \lim_{x \to 0} \dfrac{1}{x}$.

Ivo Terek
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  • Thanks for your answer, but I understand how to notate it. I'm asking for verbal terminology. –  Jun 23 '14 at 23:51
  • Oh. We talk about "lateral limits". The limit with "+" is the limit from the right, and the other is the limit from the left. – Ivo Terek Jun 23 '14 at 23:52
  • We talk about "lateral limits". The limit with "+" is the limit from the right, and the other is the limit from the left. Yes, I understand that. One could certainly describe them as "a limit that is infinite and has the same sign from both sides" versus "a limit that is infinite and has opposite signs from the two sides." However, that would be very cumbersome. –  Jun 23 '14 at 23:56