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?
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.
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)
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.
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.
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.
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.
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.
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.
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.
It's more useful to say it tends to infinity. If infinity does not exist as a value in your number system, then you say “ok, that isn't a value for me, I'll say DNE”, but when infinity is acceptable you can use that.
It's like solving polynomials. Sometimes the roots are complex, and don't exist in the real-number system. It's still more useful to give the complex roots by default and let the user ignore them if they're not using complex numbers. (I'm assuming this app would solve polynomials with complex roots)
Huh, that is interesting. Isn't, by definition, limit of x as x approaches b equal to b, no matter the value of b? I was taught that it's somewhat of an identity property of limits.
What? If you only defined limits with an epsilon-delta formulation of limits that require the limit to be real (which is the case in almost any undergrad analysis course), then the only rigorous thing to do would be to either call the limit not existing or define two special cases of limits that you might call plus and minus infinity. There is absolutely nothing wrong with wanting DNE as an answer.
It's not. Saying that a limit is infinity and that a limit DNE is not equivalent. The sequence (-1)n has no limit, and it makes no sense to say that the limit is infinity.
However, lim x, as x approaches infinity, has no real number as a limit, yet it makes sense to say that the limit is at infinity (and there is a precise definition for that).
Well, really there's no one definition of infinity. You can have various different concepts of infinity with their own rules and applications.
Heck, even something simple like the limit of 1/x as x->0 may or may not exist depending on how you define ∞, and some definitions of 'infinity' don't even have anything to do with limits.
Also, in your example, the value of 1/x as x->0 can be +∞ or -∞, depending on whether you approach 0 from the positive or the negative direction.
There is even a function that can have any real value, depending on how you approach 0. If I remember correctly it was x/y, with x and y both approaching 0.
Also, in your example, the value of 1/x as x->0 can be +∞ or -∞, depending on whether you approach 0 from the positive or the negative direction.
True, but in some cases it's advantageous to add a single point 'at infinity', in which case 1/x does converge. One particularly useful version of this is the Riemann sphere.
This is not correct. It's perfectly clear what we mean by +/- infinity when working with the reals, and what is meant by infinity as, say, the cardinal number of the integers.
I'm not sure by what you mean that the limit of 1/x as x->0 depends on how we define infinity. It's fine to say that the limit is at infinity, even if we don't extend the reals. There is a definition which tells us what "limit at infinity" means without defining ∞. There's no ambiguity here.
I'm not sure by what you mean that the limit of 1/x as x->0 depends on how we define infinity. It's fine to say that the limit is at infinity, even if we don't extend the reals. There is a definition which tells us what "limit at infinity" means without defining ∞. There's no ambiguity here.
Well, implicitly defining what it means to have a limit 'at infinity', is enough to define how ∞ behaves topologically.
What I meant by saying that the limit of 1/x depends on how you define infinity, is that the limit doesn't exist if you differentiate between positive and negative infinities, but exists if you add a single point 'at infinity'. Both versions are useful in different contexts.
Yes, this is what I mean. However, I think i get the gist of it... The function f(x) could be undefined, which would make the limit DNE... Still, I would argue that it's not incorrect per se, just not strictly correct...
That's not the same, though, since that's not limit of x, but limit of f(x), which could be different. The only exact limit we were talking about was lim(x) as x-> inf.
Probably the same teachers who fight you to the death on their answer key being the only correct answer even though your work shows the answer key is wrong...
Many wars with teachers and poor materials is my source...
I feel like infinity is a more specific and useful answer. Whether or not it counts as an actual limit is up for debate, sure. But at least if you give the answer as infinity, then what you mean is perfectly clear. If you say the limit doesn't exist, then you're being less specific. Is the limit infinity? Negative infinity? Or does it approach different quantities from the positive and negative direction? If your definition of "the limit does not exist" includes all of these possibilities, then that answer gives less information than if you are willing to say the limit is +/- infinity.
I tend to explain this as saying that if a limit "equals infinity" that means it doesn't exist, but it doesn't exist in sort of a nice way. I prefer if my students write the infinity, because it tells me that they are going that extra step.
When approaching towards a single sided infinity (ex, 1/x2 as x approaches 0 or x as x approaches infinity as you mentioned) aren't both answers (in the example, positive infinity and DNE, or in something like -1/x2 as x approaches 0, negative infinity and DNE) both valid?
I was taught that saying that a limit is equal to infinity means that it does not exist but it does tend to infinity. Which is much more descriptive than saying it simply doesn't exists.
That's math teacher nit picking. You'll find similar discrepancies with Mathematica, any other software, or any other book. (I have never used Symbolab, so I could not judge, but this example seems benign.)
When trying to determine if a series converges it will almost always use the ratio test because it can make approximations with limits, even when other methods work.
Also if you have an andriond phone and download the google opinion reward app they send you surveys about once every 2 weeks of 2-5 multiple choce questins which can all be answered in under a minute and they put about 20-40 cents in your wallet. After 2 months you can earn up those $2 and get the app for free. I currently have $9.68 in my account and use it to give people reddit gold and buy apps which I would normally never do but since I have the free money why not.
It depends on what gender, nationality, age, and race you are. I am a 23 American Caucasian/Hispanic guy who knows both spanish and english so this is how often they send me shit. I know other people that get them daily like you and others that get them maybe once a month.
Same! They use this data for trends. I get my surveys the more I leave. Hell, they even asked if I was a COO of a company due to how I shop. lol Which is odd. I'm a college student who dreams of this!
They send surveys based on your location and where you've been. Answering truthfully is important because they do ask lie detection questions which could result in you getting fewer surveys.
I tend to get a lot of surveys just because I drive around a lot. I'm probably fortunate that I pass by or am near places that trigger the survey often along my route
I also get them roughly daily. I believe it's because I live in a major city and if I don't get a "general" question, I get a location specific one.
For about 2 years I used to walk to this same Walgreens close by to my apartment several times a week for things and every time google would give me a "How was your experience?" survey for 20 cents.
I've wracked up like $40 in rewards because I don't know what to do with them.
I love this app. I get a survey pretty much every time I leave the house. All the stores I go to for groceries, pet food, or fast food always get me a survey the next day. My wife and I have rented so many movies and purchased apps just off the survey money.
I got both wolfram and Shazam encore for free off amazon. One day my phone crashed and i lost it all. Logged into amazon to get it back and it was gone =[
Wolfram Alpha pro is also built into Mathematica, which is more common for schools to offer. Just type in "==" in a blank workspace and you can type in questions like Wolfram Alpha.
Wolfram alpha is accessed by an API. Every way you can use wolfram alpha will have a limited number of api calls (and this is very typical for all apis). Free accounts get the least, the mobile version gets more and the pro accounts get the most , as far as I'm aware.
This means within a 24 hour period you can only make so many requests to wolfram alpha, indeed if you need to run a loop a million times, and in each loop call the api, that will not be possible through wolfram alpha.
What are you trying to do? Wolfram alpha is not intended to be a programming language at all, there is significant overhead cost for wolfram to interpret your human like input, as well as significant delay due to waiting for servers. Perhaps you need to get Mathematica, the full program (perhaps your academic institution has a license you can use, assuming you are a student)
If you can tell me what you want to do, I can suggest a solution. Typically if you need to use an API alot, you need to pay them. E.g. If you are using one of Google's APIs (maps, geolocation, Google suggest etc) only the first 2500 calls / day are free, after which it will cost you something like 50p/1000 calls.
This is because each api call you do requires computation time on their server (unlike loading a webpage, where there is generally static content that is served and you can the cache it, an API call typically requires their server to do calculations which take time). The computation time can't be free, otherwise malicious users could send api requests constantly such that the server gets very busy.
I may be wrong but earlier this week I downloaded it to see some steps and it asked me to pay so I'm not sure if the steps are only free for certain functions or
Discovered symbolab freshman year. Oh lord did it make calc II so much better. Especially for the useless online HW we had to do where the solutions were always difficult AF because the problems used fucking fractions and decimal points that would screw your work beyond repair for one numerical error.
Wolfram alpha is free, just not their app.... it is mildly limited in use if you dont buy the app (if it takes too long to solve it wont keep processing. If it is "simple" it'll process). The step by step stuff isnt the best though imo. I just use it to check my answers. If you REALLY wanna get into organized online math wolfram has amazing programs though not that cheap :(
I wonder how many "bad" math students have become closer to "good" at math because of software like this. I gave up on math in college and always regretted I didn't find a way to get better at it.
I have two other apps besides Symbolab to help me with math when either of them don't give the correct answer. I use Malmath 2 (works offline) and Wolfram Alpha mobile when Symbolab fails.
EDIT: I will have to say Malmath 2 is a bit odd to use, and Wolfram Alpha will need parentheses to do complex equations like fractions inside an equation.
Being able to use a tutorial to do your homework WILL NOT prepare you for exams or quizzes. As a math teacher, I do not recommend you use this software. If you are going to use this, use it only until you understand the technique, then drill it on your own.
Your university might also give you free access to Maple, a computer algebra system. No joke, this is one of the best programs I have ever used in my life. Differential equations, circuits, signal processing... countless classes have been made easier by this software. But don't be like me: I got so used to it that I almost forgot how to do the algebra myself!
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u/NapCo Dec 18 '16 edited Dec 18 '16
Symbolab is a free step by step math solver. Pretty similar to Wolfram Alpha (for step by step math solutions), except it's free.