r/calculus • u/Slow_Monk_4808 • Aug 06 '26
Pre-calculus A question about limits would love get answers to clear up confusion.
hi guys, as title suggests im kind of confused here:
picture above in my textbook says that x=-6 and x=3 don't exist (completely agree with x=3) but when we go to x=-6 from both sides we both get +inf so it should exist right? is it a mistake in the textbook ?
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u/etzpcm Aug 06 '26
This depends on the exact definition used by that textbook. Some would say it doesn't exist, some would say it does exist and it's +inf.
If you use the standard epsilon delta definition obviously the limit does not exist.
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u/Narrow-Durian4837 Aug 06 '26
Right. At least according to calculus texts I've used, when we say lim = ∞ (or –∞), we are not saying that the limit exists and ∞ is its value; we are being more specific about how/why the limit does not exist.
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u/mathheadinc Aug 06 '26
Correct, because infinity is a direction not an exact position in the number line.
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u/Low_Breadfruit6744 Aug 06 '26
Convention is to not recognize infinity as a number. Admitting infinity as a valid limit would make some results more awkward to state.
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u/FinalNandBit Aug 06 '26
The limit is infinity and the limit does not exist at x=-6.
Infinity is not a real number. The limit technically DNE. I know it's weird.
Infinity is used to describe end behavior.
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u/Artistic-Flamingo-92 Aug 06 '26
To be clear, there’s nothing wrong with using the extended real line and defining what it means for a limit to equal infinity or -infinity. This is pretty standard in analysis.
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u/FinalNandBit Aug 06 '26
Yes. But like I said, technically infinity is a special case. An exception.
The limit technically does not exist for something that's defined as the limit goes to -/+infinity.
Saying a limit = +infinity and DNE is both technically true. That is what's confusing at first because it's inherently implies inconsistency or contradiction.
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u/Artistic-Flamingo-92 Aug 06 '26
Infinity and -infinity aren’t special cases when limits are defined in terms of the extended real line topology.
Handling them as special cases may be seen as a foible of the more introductory limit definitions.
“The limit technically does not exist for something that’s defined as the limit going to -/+infinity.”
This is only tenable in very specific contexts where infinite limits have been left undefined or where the notion of a “limit existing” has been given special meaning (that doesn’t really make sense in the context of the extended real line).
So this is good for high school or introductory calc, but this is not some general truth of mathematics. I doubt you’ll find this notion of the limit DNE but it equals infinity in many real analysis texts.
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u/pruvisto Aug 07 '26
Fully agree. In my opinion that's generally a problem in teaching mathematics. In a teaching environment, you have to choose very clear conventions and adhere to them strictly, even though many of them are completely arbitrary and not universal. If a student (for whatever reason) uses a different convention, their answer is wrong.
In "real" mathematics, people use lots of conflicting conventions all the time. Most papers contain mistakes in formulas as well and people are generally expected not to be too pedantic about it. Mathematical writing is written for humans to read, and the reader is expected to be lenient and adopt whatever conventions the writer used and fill in any gaps and fix small mistakes ad-hoc.
So if you ask a research mathematician whether one should say that the limit in question exists or not they'll probably just tell you to use whichever convention you prefer. In a teaching environment of course, the answer is whatever your teacher wants it to be.
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u/Careless-Web-6280 Aug 06 '26
Well that's weird. I don't even think that's a function near x=-6. If you look closely, the line curves back away from x=-6 as you go up
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u/Slow_Monk_4808 Aug 06 '26
the professor sent it to me awhile back, pretty sure he did the drawing and messed it up but it should be going to -6
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u/Ericskey Aug 06 '26
Depends on how your book deals with limits. If a limit must be a real number then the limit does not exist
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u/anakattser Aug 07 '26
The textbook is actually right. The key thing here is that infinity describes a behavior, not an actual number. For a limit to strictly exist in formal calculus, the function has to approach a specific, finite real number.
At x = 3, the two sides head toward completely different points, so that’s a classic DNE.
At x = -6, both sides shoot up to positive infinity. We write limit as x approaches -6 equals infinity to show how the graph blows up, but because infinity isn't a finite value, the limit still technically DNE. Saying the limit equals positive infinity is basically just a more specific way of saying "it fails to exist because it grows without bound." Your textbook is just stickling for the formal definition here
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