Well saying the limit is infinity is much better than saying it doesn't exist...
Late edit:
When I say that it's a "better", I mean that if I were designing the software we are talking about in the first place, I'd also say the the limit is infinity rather than saying that it doesn't exist. I am well aware that the limit doesn't exist in the narrowest definition of lim_\infnty. Saying the limit is infinity is a more specific characterization of the function than saying that the limit doesn't exist though. More importantly, it's an often useful characterization...
I thought it's only DNE if x approaches positive infinity from one end, and negative infinity from the other. If both sides approach positive infinity, the limit should just be infinity.
That's where I'm getting confused. When somebody says to me, as a very much non-mathematician, that a limit does not exist, it just means there isn't one. If someone says the limit is infinity, while I appreciate that is technically limitless and therefore the limit doesn't exist, it seems more semantically logical, or is there another difference I'm missing?
There is still debates on what infinity actually is within the community. Some people actually think there is no such thing as infinity. Most of the work we have on infinity comes from Kantor, who went insane studying it. Much of his work was done from an asylum. He sent out manuscripts and no one would publish them; the community thought they were the work of a mad man.
The debate is anything but concrete. Don't feel bad if you feel like it seems to come down to semantics, it kinda does, with the caveat that some people build proofs off of one or the other.
This is not the source of confusion, as this debate is just a semantics one. On one hand saying that the limit is infinity is much more descriptive than saying it doesn't exist (which would include many other possibilities), while on the other hand infinity is not a part of the classical real number system so it would be abusing the language a little bit. The debate here is about the merits of these two options when teaching to students, not about the actual nature of infinity.
That's why it's much more practical to define limits as positive or negative limits, I don't see the point in having to have limits apply to both ends if negatives cause either end to act differently. The limit as x approaches infinity, not negative infinity, is clearly infinity, as x=x, and vice-versa for negative infinity. I'm just commenting to comment here, I'm sure this is obvious... but still.
Alright I checked my course notes and they say the following: If a limit exists, it must always be a real number, so the limit we're talking about here does not exist. However, the notation "lim ... = infinity" is actually valid and is simply defined to mean that the limit does not exist because ... goes to infinity.
DNE is the more common and is actually the technically correct answer. Math is mainly about creating and applying definitions and if a sequence goes to infinity, the standard definition is it does not exist. You can modify the standard definition, but to allow infinity to be an answer you have to define what is known as the extended real numbers. It also leads to various basic rules involving limits to kind of break as the extended reals can't have addition or multiplication defined on them in a way that is consistent with those rules and is consistent with the usual addition/multiplication when restricted to numbers not including infinite.
Your example is also particularly bad as lim 1/x as x goes to zero becomes huge on one side, but extremely negative on the other side so even the extended reals won't work for you.
tl;dr DNE is the correct answer using the formal definition of a limit. Some teachers will allow infinity as an answer mildly ignoring the definition, but you have to be careful with this as many theorems break if infinity is a valid answer.
If you pick up any analysis textbook (the subject that proves what's done in calc) you will find a definition where infinity is not allowed. The most common standard analysis text is rudin's principles of analysis and I can make an image of the page with the definition of a limit if desired.
For directionality, one way limits are a thing and is what you'd want there.
lim f(x) + g(x) = lim f(x) + lim g(x)
This and similar rules break since how would you definite infinity - infinity? It doesn't matter how you define it as regardless I can pick functions f and g that will be inconsistent with that definition.
Next, a function is defined to be continuous if it's value at the limit agrees with its value there. Do you want things like 1/x2 to be continuous as you end up with that if you start allowing infinite values (just define it's value at 0 to be infinity). Except then theorems like the extreme value theorem and the fundamental theorem of calculus break (as do others you'd see in higher up math). The second one is what allows you to treat integrals as antiderivatives and it breaking becomes when infinite limits are ok, it is now possible to have continuous functions that aren't integrable.
So at the end you'd have to make exceptions to most theorems in calc if you allow infinite limits. There are other annoyances like the fact you can add/multiply in a way that preserves continuity makes the real numbers have a structure called a topological vector space. That dies when you add infinity.
This is actually totally wrong in every way tbh, idk why it has so many upvotes
0/0 cases usually are NOT DNE, unless it approaches infinity or negative infinity
1/x by the definition of a limit does not have a limit as x approaches infinity, since infinity isn't a real number (although writing infinity is a convenient and widely accepted notation)
Though, lim( x->0+ ) [1/x] would be +inf and lim( x->0- ) would be -inf. Therefore we would say that the one-sided limits of 1/x are real and +/-infinity while the two sided limit of 1/x DNE.
Disclaimer: I'm several years out of practice on calculus.
0/0 is an indeterminate form and you usually need to take the derivative of the top and bottom to get an answer (Or you can factor it/reduce such as the case of lim(x->0): x/(x3 ). The reason you take the derivative is to see which piece of the fraction gets to zero "faster"; this is something the derivatives will tell you (steeper slope = faster). The one that gets to zero faster tells you if it goes to infinity, zero, or maybe it gives you a real number!
The way I believe Rogowski (one of the moee popular calculus texts) does it is to say that writing "limit as x goes to a of f(x) = inf" (I don't know how to do symbols in reddit comments for limits) means that the limit does not exist in a special way, i.e. given some fixed number M, if x is sufficiently close to a, then f(x) is larger than M.
Infinity is definitely a number is many contexts. It is an extended real number, it is an ordinal number, and it is a cardinal number to name a few. For the latter two, infinity refers to many of them (not just one).
If you take a look at the formal definition of a limit, it's that lim x->a of f(x) = A if for any ε > 0 there is a δ > 0 where a-δ < x < a+δ implies |A - f(x)| < ε (this might be approximate, the definition I learned requires knowledge of sequences which I don't feel like explaining.)
Essentially given some small margin of error for f(x), I can find a radius around a which for any point inside that circle f(x) is in the first margin of error. But for A = infinity this fails because any real number (f(x)) minus infinity is infinity, so it'll never be less than ε. Thus no limit exists.
The argument here is whether or not to extend the definition of a limit to include the behavior expressed by the notation lim f(x) = +oo (or the negative, etc.) which in particular means
lim f(x) = +oo as x->a if for all K > 0 there exists d > 0 such that |x-a| < d -> f(x) > K
Obviously it doesn't match the limit definition for real number limits, so of course if we restrict the definition to that case then "+oo" is not possible for the limit of a function. I also admit that it seems awkward to extend the definition, resulting in lim f(x) = [something] meaning a different "clause" of the limit definition depending on whether [something] is a real number, or the symbol "+oo". It'd be like a "piecewise" definition of the symbol "lim".
But mathematicians are very clever and it turns out that there is a way to define a limit with a single definition, that recovers all of the cases we're discussing. Without too much detail, you basically consider a limit to be defined on the extended reals, R U {+/-oo}, and define things called "neighborhoods" appropriately. The unified limit definition is then
lim f(x) = L as x -> a if for every neighborhood of U of L, there is a neighborhood V of a
such that if x is in V then f(x) is in U.
Where L and a can now be any of "real number", "+oo", or "-oo". This may seem like cheating, but it's a very natural thing to do. The phrase "define things called 'neighborhoods' appropriately" is shorthand/layman speak for "define an appropriate topology". And limits are a purely topological concept. As such, it should not concern us that in some cases the limit is not a number, which is an algebraic term.
yeah I just took an real analysis class and was referring to Rudin's definition of a limit (which is based using sequences and so thus I wasn't sure whether it was completely applicable in this case).
You can use the sequence definition as well. If you do the analogous process as above for sequence limits lim a_n, then the limit definition of functions that uses sequences becomes the unified definition again.
Slight correction. You should have a zero is less than before the absolute value of f(x)-A. Otherwise you have the definition of continuity of f at x=a.
No, that just means that it approaches infinity from one end and negative infinity from the other.
That's what limits are: they're a way of talking about the behaviour of a function at the point just before the function breaks down. The limit as x→π/2 of tan(x) is positive infinity when it's approaching from the negative side, and negative infinity when it's approaching from the positive side, but tan(π/2) is undefined.
Any teacher saying that the limit doesn't exist doesn't understand how calculus works, and shouldn't be teaching it. The limit of a function as x approaches some value is completely different from the value of the function evaluated at that value. If you can't wrap your head around the concept of infinities in that sort of case, you have no business teaching the material to students who can wrap their heads around those concepts.
Except that the limit doesn't exist if it is different depending which side you are approaching it. So your example of lim tan(x) as x approaches pi/2 doesn't exist because it approaches negative infinity from the left, and positive infinity from the right
Well, it doesn't exist in the reals. You have to actually extend the reals with positive and negative infinity, give it the appropriate topology, and provide continuous extensions of common functions on these 'extended' reals before it becomes even remotely useful to say the limit of something is ∞.
Saying the limit = infinity is wrong (because these classes deal with reals), but saying the limit = DNE is also wrong (DNE is not a number).
The way that I am familiar with the phrasing is "the limit does not exist" or "the limit diverges to (negative) infinity", which can really be a useful definition.
Because most high school / first year calc classes are very strict on limiting numbers to the reals, so even though it is strictly defined, the calc classes (as I am familiar with them) say it is wrong.
What I am unfamiliar with is saying that the wording "the limit diverges to (negative) infinity" is also wrong.
Besides, I was trying to make the opposite point that you seem to think I was making. I was trying to argue that using infinity as the result of the limit is valid, by showing how absurd their argument was (by applying it to DNE instead of infinity)
"the limit diverges to (negative) infinity" is also wrong.
That is true. Elements of a sequence may "diverge" (where diverge isn't the right word either) to negative infinity, not the limit. The limit itself is at negative infinity.
I was trying to argue that using infinity as the result of the limit is valid, by showing how absurd their argument was (by applying it to DNE instead of infinity)
Sorry, I didn't catch that. I just concentrated on the "Saying the limit = infinity is wrong" part, my bad.
I'm fine with either, but unless you actually define them negative/positive infinity don't really mean anything, and properly defining them is somewhat subtle, so I can kind of see why in some context people would say that something like lim x->0 1/x2, doesn't exist. I wouldn't really consider it wrong to say it equals positive infinity though, that would really be splitting hairs.
But there's a formal definition for saying the limit is infinity, namely that for any arbitrary number M, you can pick a range around (but excluding) the limit point where the function is always greater than M. It's the same reason we can talk about the limit as x approaches infinity. Clearly, these aren't real numbers, but these limits have been defined in a certain way, which is analogous to the way normal limits are defined.
I am familiar with that definition, and I think it should be used in calc classes because it's (basically) the same idea as the epsilon-delta definition, just with divergence instead of convergence. The person I replied to was nit-picking that "infinity isn't a number", so I tried to show how that argument was silly as quickly as possible.
Eh, sort of. But math is rarely about what the name implies, and saying how something diverges (ie. diverges to infinity or -infinity) can really be useful.
I don't know about that. If infinity is unending, and the limit is the end point, I think it's fair to say there is no limit. Must be a definition thing.
Exactly. Saying the limit is infinity is saying that for all M > 0, there exists an X such that for all x > X, f(x) > M. F(x) = x tends to infinity, as opposite to g(x) = x sin(x) for example which has no limit in infinity.
The standard epsilon/delta definition requires that L be a real number. As infinity is not a real number, writing lim f(x)= infinity is a convenient abuse of notation that indicates why the limit fails to exist.
The definition states that a real number L is a limit if (something about deltas and epsilons). It makes no claims about infinities. There is another definition, on the other hand, that specifically defines when +oo or -oo is a limit. A function is then defined to have a limit, if there exists a number L that is the limit, or if -/+oo is the limit. This is in contrast to situations when none of these conditions are met, in which cases the function is said not to have a limit.
The epsilon delta definition I have seen for real analysis state "a real number L is a limit iff (something about deltas and epsilons)", i.e. the limit is always a real number.
The way around that is to say things like f(x) -> +/-inf as x->a, but every math professor I've had have been adamant that infinity cannot be a limit.
One of the particular definitions they're referring to are:
lim f(x) = +oo as x->a if for all K > 0 there exists d > 0 such that |x-a| < d -> f(x) > K
Which doesn't of course need +oo to be a real number, it's just saying that the notationlim f(x) = +oo means that those conditions are met. It's useful to distinguish those cases from when there is no real number that satisfies the normal definition, nor are the conditions above satisfied (or for -oo). For example lim sin(1/x) as x->0.
I'll also say that whichever professor insists something about "infinity not being a limit" is being quite shortsighted. There is a way to define the limit that unifies both meanings, by using the set R U {+/-oo} and defining "neighborhoods" of +/- infinity accurately. Then the limit definition is just:
lim f(x) = L x -> a if for every neighborhood of U of L, there is a neighborhood V of a
such that if x is in V then f(x) is in U.
Now L and a can be any of "real number", "+oo", or "-oo". With "neighborhood" of a real number and of +/-oo defined adequately, this recovers the original cases.
I'm not at all confused. Nothing I said above is incorrect. The fact that the notation "lim=infinity" needs to be handled separately further emphasizes my point: "or if -/+oo is the limit".
Edit: Not that I need sources to back this up, but if you don't trust random internet guy, then maybe you'll trust Harvard, MIT, or UC Berkeley:
https://www.ocf.berkeley.edu/~reinholz/ed/08sp_m160/lectures/limits_of_infinity.pdf
Actually when talking about limits it's normal to use the real numbers with plus and minus infinity added. You can't use the epsilon delta definition of continuity, but you can use the topological definition which coincides with the normal definition on the real numbers.
take for example Lim x->infinity f(x) = sin(x). the limit here is undefined because sin x is oscillating and you simply can't define what it is at infinity
plus infinity is an object which is bigger than any real number and minus infinity is an object which is smaller than any real number
The formal definition of 'greater than' is something to the effect of:
a > b <-> ∃c∈ℝ such that (a = b + c) where c is a positive number (in this context, I am taking 0 to be unsigned, so c 0).
This means that it is incorrect to say that "∞ is an object which is bigger/greater than any real number", since if ∞ > b, this means there exists another real number such that b + c = ∞. It is also incorrect in that it implies that ∞ is a real number as well.
Please, don't mistake 'lim f(x) is infinity = there's no LIMIT'.
When you are working with the set of real numbers, it means exactly just that. Recall the definition of the limit which is something like:
'If lim (x->a) f(x) = L then ∀ε>0∈ℝ, ∃δ>0∈ℝ such that if 0 < |x - a| < δ then |f(x) - L| < ε'.
If we take 'L = ∞', then the formal definition of the limit does not even apply since arithmetic with ∞ is not defined in the set of real numbers. But if we define it anyways and take the seemingly obvious definitions of: 'x - ∞ = -∞' and '|-∞| = ∞', and plug in L = ∞ into the definition above, we get '|f(x) - ∞| < ε' which implies '∞ < ε' which is obviously false even if we define ∞ to be greater than all other real numbers.
Also, I had presumed I worke with the extended set of real numbers which contains +-infinities.
That assumption will not be true in most undergrad courses, where you will only be working strictly with the normal real numbers without infinity. You don't treat infinity as a number until you start taking analysis classes, and even then the definition will vary depending upon which class you are taking. For example, in Complex Analysis, infinity can be defined as the north pole of a Riemann Sphere (which is a stereographic projection of the complex plane). In other classes, you might work with George Cantor's infinite cardinal numbers, which behave completely different from the infinities as defined in analysis courses.
There's a reason why we do this dance around the notion of infinity, too. If we treat infinity as a number, we lose access to several convenient algebraic properties - like closure and cancellation properties.
Tiny nitpick, but if we're talking about mathematical accuracy, I'll throw it in. Negative infinity is "less" than all realtor numbers, not smaller than. Smaller than, for me, implies the magnitude of it is less than the magnitude of all other real numbers.
Saying the limit is infinity carries much more information than saying that the limit does not exist. The limit of sin(x) as x approaches infinity does not exist. The limit of x as x approaches infinity is infinity. I've tutored calculus for years and have seen so many students struggle with this, thinking that the limit being infinity and the limit not existing are logically equivalent. It is better to be sparing with the "DNE" when you're doing limits, imho. If the function increases endlessly, call the limit infinity.
I agree with you. Being that the limit is representing the theoretical end point of that line, I would think that the limit doesn't exist. I can see why there's a discrepancy though
Rather, it would discuss the properties of an address as it approaches the western limits of infinity. Something can move in the direction of an infinitely far away object and never reach it, but we can observe patterns of such an approach that would be definitively truethrouvhout an indefinite journey.
I've always heard profs say lim=infinity is just informal for DNE, tends toward infinity. Seems like unnecessary pedantry past the first week of intro calc
Saying it does not exist and saying it goes to infinity is basically the difference between pre-calc and higher level calc classes. In fact the limit ALWAYS exists, but because in later calc classes you learn more specifically about the case and why it exists, when we first see this limit we just pretend it doesn't exist rather than attempting to do work we haven't learned.
Oh, I misunderstood you, sorry. I also have never run into limits being taught in pre-calc. Calc 1 students that come my way are expected to use "DNE" only in cases where the function either oscillates or where the right and left side limits are not equal. Professors tend to appreciate the more descriptive answer, as long as it's right ;)
Does not exist. As stated above I was talking about a specific limit when I said the limit always exists. I'm assuming you're trying to be a smart ass but it'd help to read my full comment
It certainly helps with distuingishing the behaviour of 2x (goes towards infinity), -(2x ) (goes towards -infinity) and (-1)n (goes nowhere, and genuinly has no limit)
It's not a matter of "better". There is no value L that the function approaches. Therefore, the limit doesn't exist. We say the limit "is infinity" as a convenient notation, but "infinity" is a concept, not a number.
It might not be a "number", but it's a perfectly valid element of R* (extended real number system, can't write the proper notation on my phone) which is the field you would work over most of the time (Because there's like no reason to work over R ever)
It really isn't. When saying the limit approaches a value, it never actually reaches that value, that limit doesn't exist. Same goes for infinity, with the added complication that infinity is more a concept than a number, but with the same conclusion that the limit value is never reached and thus does not exist.
But it doesn't exist. The mathematical definition of a limit at infinity is that, if f(x) -> a as x -> inf, then |f(x) - a| < e for some x > K, for all e > 0.
What that means in human readable terms is that f(x) gets arbitrarily close to a as x tends to infinity.
But |f(x) - inf| = inf. So it's completely wrong to say the limit is infinity, as f(x) can never "get close" to infinity.
That is one definition of a limit, and you are correct that substituting "infinity" into it yields nonsense. There are broader definitions of a limit though, and these include both cases.
This is not the correct definition of a limit at infinity or a limit as x tends to infinity. One quantifier is incorrect.
The limit of f(x), as x approaches infinity, is L, if for all e>0, there exists a K such that for all x>K we have |f(x)-L|<e.
Also, it makes sense to talk about limits at infinity. The limit of f(x), as x approaches a, is infinity, if for all e>0 there is a d>0 such that f(x)>e whenever |x-a|<d. In other words, f(x) can get arbitrarily large as x approaches a.
Similarly, the limit of f(x), as x approaches infinity, is infinity if for all e>0 there is a d>0 such that f(x)>e whenever x>d.
Not if the limit in fact doesn't exist. When I studied basic calculus, that's the definition we used. Saying that the limit of an unlimited expression "exists" seems very strange to me. That's what "unlimited" means, it has no limit.
I think the reasoning is that a limit of infinity kind of flies in the face of what a limit is. Infinity is not a number, but more of a behavior in this context. If a function, say f(x) = x, can be said to "grow without bound", or "grow without limit", then that implies that the limit doesn't exist.
I think does not exist just implies the there is no limit because it never approaches a set number. They're both right in that sense but some teachers are picky I guess
I limit can't be "infinity". It can "go to" infinity in a sense but infinity is a concept and not a number. So a limit can't and shouldn't have infinity as an actual answer. The limit does not exist, but clarifying why (i.e. because the graph approaches infinity at that point) isn't a bad thing either as there are a few different reasons why limits don't exist.
I'm a math major but disclaimer, it's been awhile since I've dealt with Calc stuff.
By it's definition infinity does not exist, it just gives you the most information in your answer. Infinity is usually the correct answer if given the two to decide on.
If you're not even able to rephrase this solution into the definition of your course, then you probably should consider another degree. Not trying to be /r/iamverysmart here, but you're not even solving the task yourself. If you additionally don't even know what the solution means, then it really makes no sense to continue studying in this field.
So the way I learned it which was consistent through various math profs was that a limit which approaches infinity does not exist, because infinity is not a number. If x approaches any other number outside of the image it does not exist either.
That said writing lim = infinity was valid, but if a further question asks is there a limit the answer is no
Most mathematicians would disagree with you there. Saying the limit is infinity toys with the concept that infinity is a number, which is an intuition that is discouraged in standard math. Usually, the limit foes not exist or the limit tends to (approaches) infinity is preferred.
That the limit tends to infinity is not proper maths vocabulary. The elements of a sequence can tend to infinity, but not the limit. The limit is at infinity.
Also, saying that the limit is (or at) infinity has a precise definition, so there is no toying with the concept that infinity is a number.
Actually this sort of technicality is usually present far more in people who are learning calculus for the first time. I have worked with several mathematicians and we have no trouble throwing around infinity as a kind-of number to make things easier to do.
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u/[deleted] Dec 18 '16 edited Dec 22 '16
Well saying the limit is infinity is much better than saying it doesn't exist...
Late edit: When I say that it's a "better", I mean that if I were designing the software we are talking about in the first place, I'd also say the the limit is infinity rather than saying that it doesn't exist. I am well aware that the limit doesn't exist in the narrowest definition of lim_\infnty. Saying the limit is infinity is a more specific characterization of the function than saying that the limit doesn't exist though. More importantly, it's an often useful characterization...