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If the left and right limits of a function $f$ differ, do we then say that "$f$ has no limit at $c$"? (This sounds wrong considering that f has both left and right limits... but we obviously can't claim that $f$ has two limits either, since limits are unique.) Or is it simply not meaningful to be talking about the limit of $f$ at $c$ (because that would imply that the left and right limits at c are identical)?

(I do understand the concept, but linguistically, it's akin to saying that I have a left hand, and I also have a right hand, but I don't necessarily have a hand.)

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(This question was recently edited. I didn't see the question when it was originally asked.) The terms unilateral limit (or one-sided limit) and bilateral limit (or two-sided limit) are often used, and for the situation you bring up one would say that both unilateral limits exist and the bilateral limit does not exist. – Dave L. Renfro Feb 1 '13 at 15:47
@DaveL.Renfro Thank you for introducing the terms! But that doesn't negate the above linguistic conundrum. I would explain it as being due to the fact that while the "left" in "left hand" is sort of like a nonrestrive 'clause' that expands on the meaning of the phrase "left hand", the "left" in "left limit" is sort of like a restrictive 'clause' that qualifies the meaning of the phrase "left limit". Or to put it another way: "I have a hand, and it is left." Whereas "I have a left-limit, but it's not necessarily a limit per se." Perhaps adding the hyphen in such a case is a good idea! – Ryan Feb 1 '13 at 21:04
I was in kind of a hurry this morning when I read your question and wrote my comment, and I apparently didn't internalize very well your "left hand" and "right hand" comment, since the analogy now seems to me to be misdirected (as I think you're also thinking). You might find the notion of a cluster set at a point of interest. I posted a short expository essay about it in this 26 January 2007 Math Forum AP-Calculus post. – Dave L. Renfro Feb 1 '13 at 21:07
@DaveL.Renfro My left-hand-right-hand example was deliberately absurd; the point being: if "left limit" and "left hand" aren't linguistically analogous, it sure isn't clear off the bat! p.s. Thank you for the link. – Ryan Feb 1 '13 at 23:38
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In that case the limit does not exist (which is the same as saying that the limit is non-existent). The reason is that limit has a precise meaning and one-sided limit also has a precise meaning. A function having a one-sided limit, or even both one-sided different limits does not qualify as f having half of its limit. The definitions are clear as are the implications: If the limit exist then the one-sides limits exist as well and are equal to the limit. Consequently, if the one-sided limits exist and are different then the limit does not exist.

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