The other comments are correct: is undefined. Similarly, the limit of as approaches is also undefined. However, if you take the limit of as approaches zero from the left or from the right, you get negative and positive infinity respectively.

Answer from Ethan Brown on Stack Exchange
🌐
Reddit
reddit.com › r/learnmath › why do we say 1/0=undefined instead of 1/0=infinity?
r/learnmath on Reddit: Why do we say 1/0=undefined instead of 1/0=infinity?
October 24, 2020 -

Like 10/2- imagine a 10 square foot box, saying 10 divided by 2 is like saying “how many 2 square foot boxes fit in this 10 square foot box?” So the answer is 5.

But if you take the same box and ask “how many boxes that are infinitely small, or zero feet squared, can fit in the same box the answer would be infinity not “undefined”. So 10/0=infinity.

I understand why 2/0 can’t be 0 not only because that doesn’t make and since but also because it could cause terrible contradictions like 1=2 and such.

Ah math is so cool. I love infinity so if anyone wants to talk about it drop a comment.

Edit: thanks everyone so much for the answers. Keep leaving comments though because I’m really enjoying seeing it explained in different ways. Also it doesn’t seem like anyone else has ever been confused by this judging by the comment but if anyone is I really liked this video https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:division-zero/v/why-dividing-by-zero-is-undefined

Top answer
1 of 5
7
Because a/b = x which means a = bx. For example : 6/2 = 3 which is same as saying 6 = 2*3. So when we divide 1 (or any number) by 0, it implies that 1/0 = x i.e. 1 = 0*x , which we know is not possible as anything multiplied by 0 gives you 0. Therefore we say it is undefined and not infinity.
2 of 5
2
Usually it's because infinity isn't an element of the set the division is defined on. If you do include infinity with the reals then there are two common ways to do it. The Affinely Extended Reals, where positive infinity and negative infinity are included. In this case if we allow 1/0=infinity, then 1/(-0) presents a problem since1/(-0)=-1/0=-infinity would be invalid. Alternatively there is the Projectively Extended Reals, where we want 1/0=infinity, so we accept that infinity=-infinity to allow 1/(-0)=1/0. This is equivalent to a number circle, where traveling infinitely far in either the positive or negative direction leads us to the same point.
🌐
Stack Exchange
math.stackexchange.com › questions › 127376 › i-have-learned-that-1-0-is-infinity-why-isnt-it-minus-infinity
indeterminate forms - I have learned that 1/0 is infinity, why isn't it minus infinity? - Mathematics Stack Exchange
Top answer
1 of 13
93

The other comments are correct: is undefined. Similarly, the limit of as approaches is also undefined. However, if you take the limit of as approaches zero from the left or from the right, you get negative and positive infinity respectively.

2 of 13
89

does tend to as you approach zero from the left, and as you approach from the right:

That these limits are not equal is why is undefined.

Videos
10:08
🌐 YouTube
Can I ask why 1 divided by 0 doesn’t equal infinity? - YouTube
August 29, 2025
02:00
🌐 YouTube
1/∞ = 0 Proof - YouTube
April 20, 2025
05:36
🌐 YouTube
1/0 = infinite is explained | Breaking the rules of Mathematics.
March 19, 2023
09:53
🌐 YouTube
Why 1/0 is infinity ? Concept of Limit - YouTube
March 30, 2022
03:41
🌐 YouTube
1/0 = Undefined or Infinity: Easy proof to understand with a real ...
June 24, 2018
🌐 youtube.com
Is (1/0) Infinity - Limits and Derivatives | Class 11 Maths - YouTube
May 26, 2020
View all
View all
🌐
Quora
quora.com › If-1-0-is-infinity-then-what-is-1-0
If 1/0 is infinity then what is -1/0? - Quora
Answer (1 of 11): If you define some number structure where 1/0 = ∞, then, if it’s going to be as nicely behaved as possible, -1/0 = ∞ too. Shouldn’t it be -∞? Yes, it should—but -∞ should equal ∞, if ∞ is defined as 1/0. Think ...
🌐
MathsisFun
mathsisfun.com › calculus › limits-infinity.html
Limits to Infinity
We have seen two examples, one went to 0, the other went to infinity. In fact many infinite limits are actually quite easy to work out, when we figure out "which way it is going", like this: Functions like 1/x approach 0 as x approaches infinity.
🌐
Physics Forums
physicsforums.com › mathematics › calculus
Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1? • Physics Forums
December 20, 2021 - The discussion also reflects varying ... = c, then a = b*c and b = a/c Therefore if 1/ ∞ = 0, ∞ * 0 should be equal to 1 and 1/0 = ∞...
🌐
Math Central
mathcentral.uregina.ca › qq › database › qq.02.06 › evan1.html
1/infinity and 1/0
Question: I was thinking the other day when i was in math class that when you divide 1 by say n you'll get 1/n. As the value of n increases the smaller the number you get. So if you divide 1/infinity would that equal zero? And if that is true then would 1/0=infinity be true also · Your observation ...
🌐
Wikipedia
en.wikipedia.org › wiki › Indeterminate_form
Indeterminate form - Wikipedia
December 30, 2025 - In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated. ... {\displaystyle \textstyle \lim {\frac {\beta }{\alpha }}=1} , they are called equivalent infinitesimal (equiv. ... {\displaystyle \lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta \beta '\alpha '}{\beta '\alpha '\alpha }}=\lim {\frac {\beta }{\beta '}}\lim {\frac {\alpha '}{\alpha }}\lim {\frac {\beta '}{\alpha '}}=\lim {\frac {\beta '}{\alpha '}}} ... {\displaystyle 0/0} , one can make use of the following facts about equivalent infinitesimals (e.g.,
Some examples and non-examplesEvaluating indeterminate formsList of indeterminate forms
🌐
Quora
quora.com › Is-1-0-infinity
Is 1/0 infinity? - Quora
Answer (1 of 266): Technically, it is undefined. However, if one were to take the limit of 1/x as x approaches 0 from the positive end, it would tend to positive infinity. Going from the negative end, it would be negative infinity.
Find elsewhere
🌐
Narkive
mathematics.science.narkive.com › kErfp69Q › why-1-0-is-infinity-please-prove
Why 1/0 is infinity ???????? Please prove?
So if you try puting it any value ... definition. thus ∞ is mathematically not defined (and hence no algebra rules on this value works). so 1/0 = ∞ = not defined = infinity ( all different ways to say the same thing) ......
🌐
Stack Overflow
stackoverflow.com › questions › 53129908 › why-is-1-0-infinity-and-1-0-infinity
Why is 1/0=Infinity and 1/-0=-Infinity
Top answer
1 of 1
6

This is because the IEEE 754 specifications define it like that.

There is however a reasoning for this: the affinely extended real number system extends the real numbers with the two infinities, which gives some more room for calculating with limits. So with this extension a division by zero is not undefined or NaN.

Consider that the following is true for positive x:

      limx→0(x) = limx→0(-x)

However the following is not true for positive x:

      limx→0(1/x) = limx→0(1/-x)

Note how the above comparisons with limit notation map to the comparisons you listed:

0 === -0;// true
1/0 === 1/-0;// false

Secondly, a division always maintains the following invariance: the result is negative if and only when exactly one of the operands is negative.

Both of these considerations give some credence as to why in IEEE 754:

1/0 === Infinity
1/-0 === -Infinity
🌐
Wikipedia
en.wikipedia.org › wiki › Division_by_zero
Division by zero - Wikipedia
1 week ago - Since any number multiplied by 0 is 0, the expression ⁠ ... Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type of mathematical singularity.
Elementary arithmeticEarly attemptsCalculusAlternative number systemsHigher mathematicsComputer arithmeticHistorical accidentsSourcesFurther reading
🌐
Reddit
reddit.com › r/learnmath › why does 1/infinity = 0 rather than 0.0 repeating leading to 1?
r/learnmath on Reddit: why does 1/infinity = 0 rather than 0.0 repeating leading to 1?
June 3, 2024 -

sorry if the question doesnt make sense i havent been invested in math theory for long as ive only taken alg 2 and minor precalc but why is it that one over infinity equals zero rather than an infinitely small finite number? from my thoughts i feel as if it cant be zero because if you have anumerator there is a value no matter the size of a denominator, almost like an asymptotic relationship with the value reaching closer to zero but never hitting it. i understand zero is a concept so you cant operate with it so you cant exactly create a proof algebraicly but then how could you know it equals zero? just need second thoughts as its a comment debate between me and my brother. many thanks!

edit: my bad i wasnt very misunderstood on alot of things and the question was pretty dumb in hindsight, my apologies

Top answer
1 of 35
108
If the zeros are repeating there is no end, therefore there is no 1 at the end because there is no end to begin with
2 of 35
32
Part of the problem is that 1/infinity is actually very hard to define, because "let's treat infinity as a number!" usually leads to strange situations. Like for example, what's infinity + 1? Is it still infinity? Does that mean, if we "subtract infinity" from both sides, that 1 = 0? What is infinity - infinity, anyway? One way people get around this is to use limits. Say, instead of 1 / infinity, you have 1/x. This is a function. Then we might (depending on the function) be able to ask, "what happens as we keep increasing the value of x?" People phrase this sometimes as "what happens when x 'goes to infinity'?", but what they really mean is, "what happens when we make x an arbitrarily large number, and then an even larger one, and so on?" Then people ask, "as x gets larger and larger, does the value of 1/x approach anything in particular? Does it become closer and closer to some exact value? Does it converge?" Answering this kind of question is actually much more doable than trying to figure out "what is 1 / infinity?" because, instead of having to figure out how to do division by infinity (something which does not really have a concrete definition), we're just dealing with a finite number, x. To answer the question though, what we see as x gets bigger and bigger is that 1/x does actually converge on one specific value! That value is 0. Note that 1/x is never actually equal to 0 — it just approaches it. It gets closer, and closer, and closer. People do sometimes write this as "1 / infinity = 0". But I think it would be fair to say that, when people write it this way, that can be... very misleading, depending on the reader.
🌐
Math Forums
mathforums.com › home › mathematics › general math
1/0=Infinity and Negative Infinity. Change My Mind. | Math Forums
October 22, 2023 - We use infinity essentially to describe when this happens, not to state it is a number. We use it in numbers sometimes to show this kind of upper and lower bounds in numbers, but not because it is a number itself. This is not proper usage, but say for example how 1/10^∞ would become negligible after taking limits, thus becomes 0 in whatever equation you're taking limits in.
🌐
Cut the Knot
cut-the-knot.org › WhatIs › Infinity › BigNumber.shtml
Infinity As a Limit
In the real number system, 1/0 is quite meaningless, or, at best, ambiguous. Limits are studied at the beginning Calculus courses where it is shown that if ... This tells us that the expression 0·∞ will forever remain undefined. ... As with the product, it is not always possible to use that ...
🌐
Reddit
reddit.com › r/learnmath › is 0/0 = infinite ?
r/learnmath on Reddit: is 0/0 = infinite ?
January 6, 2021 -

I saw too many people saying this statement is true but its not I can prove it. Sometimes you find the limit of a function with an "a" integer 0/0 so you change the function form and make with a known form and limit of a/0+ is infinite there cant be 0/0 is infinite ..

Top answer
1 of 6
32
0/0 is not defined. That's all. Now you could have two functions f(x) and g(x) such as their limite is equal to 0 when x is close to 0, and you're trying to look if f(x)/g(x) is defined for x close to 0. For example I can take f(x) = x and g(x) = x , I'll have f(0)=0 and g(0)=0 however if I calculate for any x different of 0 f(x)/g(x) = 1 , for x=0 it's still undefined but the limit of f(x)/g(x) as x goes close to 0 is 1. Now could do the same with f(x) =2x and g(x) = x. The limit at 0 of f(x)/g(x) = 2. In fact you could do this with any number a and have f(x)=ax and g(x) = x, the limit would be equal to a. But there are also tons of example where the limits is not finite f(x) = x and g(x) = x² then f(x)/g(x) = 1/x doesn't have any limit at 0. Conclusion : if you have a 0/0, it will always depends on the function you study, there are no general solutions to this limit
2 of 6
5
It's either undefined or indeterminate, depending on the circumstances.
🌐
The Hindu
thehindu.com › opinion › letters › infinity-undefined › article3863329.ece
Infinity & undefined - The Hindu
September 5, 2012 - In mathematics, expressions like 1/0 are undefined. But the limit of the expression 1/x as x tends to zero is infinity. Similarly, expressions like 0/0 are undefined. But the limit of some expressions may take such forms when the variable takes ...
🌐
Physics Forums
physicsforums.com › mathematics › general math
Disputing 1 / 0 = Infinity: Agree or Disagree? • Physics Forums
January 11, 2024 - I would sooner assume that they're ... is understood ("In the extended reals, is 1/0 infinite?"), then the answer is no: it is undefined....
🌐
Physics Forums
physicsforums.com › science education and careers › stem educators and teaching
Does limit 1/x at zero equal infinity? How it is accepted in High School now? • Physics Forums
October 9, 2023 - Per what my son says, that limit should be considered as undefined because ##\lim_{x \to 0^+}\frac{1}{x}=+\infty## and ##\lim_{x \to 0^-}\frac{1}{x}=-\infty## and as these infinities have different signs, the general limit does not exist (even ...
🌐
Medium
prabhatmahato.medium.com › why-is-any-number-over-0-undefined-or-what-we-say-infinity-5318dc5b0153
Why is any number over 0 undefined or what we say infinity? | by Prabhat Mahato | Medium
March 31, 2023 - Uh Oh, it looks like this will ... to be subtracted from 1 for the result to be 0. So simply, we cannot define 1/0 in normal division terms and hence is undefined....
🌐
Reddit
reddit.com › r/learnmath › can i assume 1/0 as infinity whilst dealing with limits?
r/learnmath on Reddit: Can I assume 1/0 as infinity whilst dealing with limits?
March 1, 2024 -

So I was solving some calculus questions primarily dealing with lhôpital rule

Here is the question

lim x tends to 0 (1+sinx)^1/x²

Now what I did was to take the natural log of the function as we know that e^lnf(x)=e^L

A1: 1/x² ln(1+sinx)

A2: ln(1+sinx)/x²

A3:Applying the lhôpital rule

1/(1+sinx)/2x (note I am neglecting the derivative of (1+sinx) which we will get)

Now here I am having question

Can I substitute x=0 and say infinity?

Then prove that e^infinity = undefined and thus limit DNE?

or should I just stop at A3 step?

Or did I just get entire question wrong?

S

ol

Top answer
1 of 2
4
More or less yes, but you do need to be careful of sign. The function 1/x at 0 diverges to +∞ from the right and -∞ from the left, so the limit is not unique, i.e. DNE. But the function 1/x2 diverges to +∞ from both the right and left so one can confidently say the limit is +∞. Whether you want to call this a DNE limit or an ∞ limit seems to depend on the context. Personally, as a topologist I prefer making the distinction between diverging due to having multiple limit points and diverging due to “going outside the space”. (For those of you who may want to question me, the concept is important in understanding compactifications.) As for the actual problem you are working on, that is a case where you have ±∞ both as limit points. So no this limit does not exist.
2 of 2
2
A few questions/comments: Why did you neglect the derivative? You cannot substitute x=0 with the limit. You take the limit as x approaches 0. Consider the right-hand and left-hand limits separately. The limit as x approaches infinity of ex is infinity, not undefined. I suppose you could treat infinity as undefined, but usually it’s infinity. Also, einfinity is not allowed.