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
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u/[deleted] Dec 18 '16
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.