An Intuitive Introduction To Limits

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Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them:

  • What is a limit? Our best prediction of a point we didn’t observe.
  • How do we make a prediction? Zoom into the neighboring points. If our prediction is always in-between neighboring points, no matter how much we zoom, that’s our estimate.
  • Why do we need limits? Math has “black hole” scenarios (dividing by zero, going to infinity), and limits give us a reasonable estimate.
  • How do we know we’re right? We don’t. Our prediction, the limit, isn’t required to match reality. But for most natural phenomena, it sure seems to.

Limits let us ask “What if?”. If we can directly observe a function at a value (like x=0, or x growing infinitely), we don’t need a prediction. The limit wonders, “If you can see everything except a single value, what do you think is there?”.

When our prediction is consistent and improves the closer we look, we feel confident in it. And if the function behaves smoothly, like most real-world functions do, the limit is where the missing point must be.

Key Analogy: Predicting A Soccer Ball

Pretend you’re watching a soccer game. Unfortunately, the connection is choppy:

soccer limits

Ack! We missed what happened at 4:00. Even so, what’s your prediction for the ball’s position?

Easy. Just grab the neighboring instants (3:59 and 4:01) and predict the ball to be somewhere in-between.

And… it works! Real-world objects don’t teleport; they move through intermediate positions along their path from A to B. Our prediction is “At 4:00, the ball was between its position at 3:59 and 4:01″. Not bad.

With a slow-motion camera, we might even say “At 4:00, the ball was between its positions at 3:59.999 and 4:00.001″.

Our prediction is feeling solid. Can we articulate why?

  • The predictions agree at increasing zoom levels. Imagine the 3:59-4:01 range was 9.9-10.1 meters, but after zooming into 3:59.999-4:00.001, the range widened to 9-12 meters. Uh oh! Zooming should narrow our estimate, not make it worse! Not every zoom level needs to be accurate (imagine seeing the game every 5 minutes), but to feel confident, there must be some threshold where subsequent zooms only strengthen our range estimate.

  • The before-and-after agree. Imagine at 3:59 the ball was at 10 meters, rolling right, and at 4:01 it was at 50 meters, rolling left. What happened? We had a sudden jump (a camera change?) and now we can’t pin down the ball’s position. Which one had the ball at 4:00? This ambiguity shatters our ability to make a confident prediction.

With these requirements in place, we might say “At 4:00, the ball was at 10 meters. This estimate is confirmed by our initial zoom (3:59-4:01, which estimates 9.9 to 10.1 meters) and the following one (3:59.999-4:00.001, which estimates 9.999 to 10.001 meters)”.

Limits are a strategy for making confident predictions.

Exploring The Intuition

Let’s not bring out the math definitions just yet. What things, in the real world, do we want an accurate prediction for but can’t easily measure?

What’s the circumference of a circle?

Finding pi “experimentally” is tough: bust out a string and a ruler?

We can’t measure a shape with seemingly infinite sides, but we can wonder “Is there a predicted value for pi that is always accurate as we keep increasing the sides?”

Archimedes figured out that pi had a range of

\displaystyle{3 \frac{10}{71} < \pi < 3 \frac{1}{7} }

using a process like this:

It was the precursor to calculus: he determined that pi was a number that stayed between his ever-shrinking boundaries. Nowadays, we have modern limit definitions of pi.

What does perfectly continuous growth look like?

e, one of my favorite numbers, can be defined like this:

\displaystyle{e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n}

We can’t easily measure the result of infinitely-compounded growth. But, if we could make a prediction, is there a single rate that is ever-accurate? It seems to be around 2.71828…

Can we use simple shapes to measure complex ones?

Circles and curves are tough to measure, but rectangles are easy. If we could use an infinite number of rectangles to simulate curved area, can we get a result that withstands infinite scrutiny? (Maybe we can find the area of a circle.)

Can we find the speed at an instant?

Speed is funny: it needs a before-and-after measurement (distance traveled / time taken), but can’t we have a speed at individual instants? Hrm.

Limits help answer this conundrum: predict your speed when traveling to a neighboring instant. Then ask the “impossible question”: what’s your predicted speed when the gap to the neighboring instant is zero?

Note: The limit isn’t a magic cure-all. We can’t assume one exists, and there may not be an answer to every question. For example: Is the number of integers even or odd? The quantity is infinite, and neither the “even” nor “odd” prediction stays accurate as we count higher. No well-supported prediction exists.

For pi, e, and the foundations of calculus, smart minds did the proofs to determine that “Yes, our predicted values get more accurate the closer we look.” Now I see why limits are so important: they’re a stamp of approval on our predictions.

The Math: The Formal Definition Of A Limit

Limits are well-supported predictions. Here’s the official definition:

\displaystyle{ \lim_{x \to c}f(x) = L } means for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε

Let’s make this readable:

Math EnglishHuman English
\displaystyle{ \lim_{x \to c}f(x) = L }
  means
When we “strongly predict” that f(c) = L, we mean
for all real ε > 0for any error margin we want (+/- .1 meters)
there exists a real δ > 0there is a zoom level (+/- .1 seconds)
such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < εwhere the prediction stays accurate to within the error margin

There’s a few subtleties here:

  • The zoom level (delta, δ) is the function input, i.e. the time in the video
  • The error margin (epsilon, ε) is the most the function output (the ball’s position) can differ from our prediction throughout the entire zoom level
  • The absolute value condition (0 < |x − c| < δ) means positive and negative offsets must work, and we’re skipping the black hole itself (when |x – c| = 0).

We can’t evaluate the black hole input, but we can say “Except for the missing point, the entire zoom level confirms the prediction f(c) = L.” And because f(c) = L holds for any error margin we can find, we feel confident.

Could we have multiple predictions? Imagine we predicted L1 and L2 for f(c). There’s some difference between them (call it .1), therefore there’s some error margin (.01) that would reveal the more accurate one. Every function output in the range can’t be within .01 of both predictions. We either have a single, infinitely-accurate prediction, or we don’t.

Yes, we can get cute and ask for the “left hand limit” (prediction from before the event) and the “right hand limit” (prediction from after the event), but we only have a real limit when they agree.

A function is continuous when it always matches the predicted value (and discontinuous if not):

\displaystyle{\lim_{x \to c}{f(x)} = f(c)}

Calculus typically studies continuous functions, playing the game “We’re making predictions, but only because we know they’ll be correct.”

The Math: Showing The Limit Exists

We have the requirements for a solid prediction. Questions asking you to “Prove the limit exists” ask you to justify your estimate.

For example: Prove the limit at x=2 exists for

\displaystyle{f(x) = \frac{(2x+1)(x-2)}{(x - 2)}}

The first check: do we even need a limit? Unfortunately, we do: just plugging in “x=2″ means we have a division by zero. Drats.

But intuitively, we see the same “zero” (x – 2) could be cancelled from the top and bottom. Here’s how to dance this dangerous tango:

  • Assume x is anywhere except 2 (It must be! We’re making a prediction from the outside.)
  • We can then cancel (x – 2) from the top and bottom, since it isn’t zero.
  • We’re left with f(x) = 2x + 1. This function can be used outside the black hole.
  • What does this simpler function predict? That f(2) = 2*2 + 1 = 5.

So f(2) = 5 is our prediction. But did you see the sneakiness? We pretended x wasn’t 2 [to divide out (x-2)], then plugged in 2 after that troublesome item was gone! Think of it this way: we used the simple behavior from outside the event to predict the gnarly behavior at the event.

We can prove these shenanigans give a solid prediction, and that f(2) = 5 is infinitely accurate.

For any accuracy threshold (ε), we need to find the “zoom range” (δ) where we stay within the given accuracy. For example, can we keep the estimate between +/- 1.0?

Sure. We need to find out where

\displaystyle{|f(x) - 5| < 1.0}

so


\begin{align*}
|2x + 1 - 5| &< 1.0 \\
|2x - 4| &< 1.0 \\
|2(x - 2)| &< 1.0 \\
2|(x - 2)| &< 1.0 \\
|x - 2| &< 0.5
\end{align*}

In other words, x must stay within 0.5 of 2 to maintain the initial accuracy requirement of 1.0. Indeed, when x is between 1.5 and 2.5, f(x) goes from f(1.5) = 4 to and f(2.5) = 6, staying +/- 1.0 from our predicted value of 5.

We can generalize to any error tolerance (ε) by plugging it in for 1.0 above. We get:

\displaystyle{|x - 2| < 0.5 \cdot \epsilon}

If our zoom level is “δ = 0.5 * ε”, we’ll stay within the original error. If our error is 1.0 we need to zoom to .5; if it’s 0.1, we need to zoom to 0.05.

This simple function was a convenient example. The idea is to start with the initial constraint (|f(x) – L| < ε), plug in f(x) and L, and solve for the distance away from the black-hole point (|x – c| < ?). It’s often an exercise in algebra.

Sometimes you’re asked to simply find the limit (plug in 2 and get f(2) = 5), other times you’re asked to prove a limit exists, i.e. crank through the epsilon-delta algebra.

Flipping Zero and Infinity

Infinity, when used in a limit, means “grows without stopping”. The symbol ∞ is no more a number than the sentence “grows without stopping” or “my supply of underpants is dwindling”. They are concepts, not numbers (for our level of math, Aleph me alone).

When using ∞ in a limit, we’re asking: “As x grows without stopping, can we make a prediction that remains accurate?”. If there is a limit, it means the predicted value is always confirmed, no matter how far out we look.

But, I still don’t like infinity because I can’t see it. But I can see zero. With limits, you can rewrite

\displaystyle{\lim_{x \to \infty}}

as

\displaystyle{\lim_{\frac{1}{x} \to 0}}

You can get sneaky and define y = 1/x, replace items in your formula, and then use

\displaystyle{\lim_{y \to 0^+}}

so it looks like a normal problem again! (Note from Tim in the comments: the limit is coming from the right, since x was going to positive infinity). I prefer this arrangement, because I can see the location we’re narrowing in on (we’re always running out of paper when charting the infinite version).

Why Aren’t Limits Used More Often?

Imagine a kid who figured out that “Putting a zero on the end” made a number 10x larger. Have 5? Write down “5″ then “0″ or 50. Have 100? Make it 1000. And so on.

He didn’t figure out why multiplication works, why this rule is justified… but, you’ve gotta admit, he sure can multiply by 10. Sure, there are some edge cases (Would 0 become “00″?), but it works pretty well.

The rules of calculus were discovered informally (by modern standards). Newton deduced that “The derivative of x^3 is 3x^2″ without rigorous justification. Yet engines whirl and airplanes fly based on his unofficial results.

The calculus pedagogy mistake is creating a roadblock like “You must know Limits™ before appreciating calculus”, when it’s clear the inventors of calculus didn’t. I’d prefer this progression:

  • Calculus asks seemingly impossible questions: When can rectangles measure a curve? Can we detect instantaneous change?
  • Limits give a strategy for answering “impossible” questions (“If you can make a prediction that withstands infinite scrutiny, we’ll say it’s ok.”)
  • They’re a great tag-team: Calculus explores, limits verify. We memorize shortcuts for the results we verified with limits (d/dx x^3 = 3x^2), just like we memorize shortcuts for the rules we verified with multiplication (adding a zero means times 10). But it’s still nice to know why the shortcuts are justified.

Limits aren’t the only tool for checking the answers to impossible questions; infinitesimals work too. The key is understanding what we’re trying to predict, then learning the rules of making predictions.

Happy math.

Other Posts In This Series

  1. A Gentle Introduction To Learning Calculus
  2. How To Understand Derivatives: The Product, Power & Chain Rules
  3. How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
  4. An Intuitive Introduction To Limits
  5. Why Do We Need Limits and Infinitesimals?
  6. Learning Calculus: Overcoming Our Artificial Need for Precision
  7. Prehistoric Calculus: Discovering Pi
  8. A Calculus Analogy: Integrals as Multiplication
  9. Calculus: Building Intuition for the Derivative
  10. Understanding Calculus With A Bank Account Metaphor
  11. A Friendly Chat About Whether 0.999... = 1
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31 Comments

  1. Indeed, one of the great tragedies of mathematical education is that we teach calculus backwards. The epsilon-delta business of Cauchy and Weirestrass is, of course, key in the field of analysis. But high school and university students are there to learn calculus, not calculus of variations, right? For 150 years, we did quite well sticking with Liebniz’s notion of infinitesimal quantities, a concept that’s all but disappeared from modern calculus courses. (I hadn’t heard of an ‘infinitesimal’ until I stumbled upon this site in the midst of my high school calculus course).

    Anyway, keep up the good work, Kalid. Another excellent article.

  2. I love this website & its emails & how through this site I confirm that there are other people out there that find math to be magical.

  3. Hello, great post as always! May I ask a question? About limits in the indeterminate form 0/0, I can’t understand why algebraic manipulation works! Any insight will be welcomed! Thanks,

    Liana

  4. @Joe: Great point, thanks for the comment. Exactly, we teach high school calculus as if we’re hard-nosed theoreticians interested in the mechanics of how calculus is put together (a bit like learning organic chemistry to see how gasoline is combusted before taking driver’s ed). Happy you enjoyed the article.

    @Sean: Really appreciate it!

    @Liana: Great question. I’d like to do a follow-up on some of the subtleties about how to resolve indeterminate forms. In this example, my intuition is the points *outside* the black hole do not have any issue with (x – 2) [for example], so can divide it out easily. And we are actually using the surrounding points, not the the “black hole” itself, to make the estimate.

  5. this was dynamite !
    “The calculus pedagogy mistake is creating a roadblock like “You must know Limits™ before appreciating calculus”, when it’s clear the inventors of calculus didn’t”

    loved it. thanks a ton. when i was in junior school [1977], my village school teacher used a rope and sticks to explain pi. it used to be called ‘sulba sutra’ [string [as in rope and string] principles] in ancient india. indians didnt give much importance to rigorous proof. if something could be directly measured [like d/dt [x square] = 2x] then that was it!

    i wish more teachers would tech more ‘practical’ maths.

    string theory is my fav example of useless maths – 30 yrs of research funding sunk, without a SINGLE falsi-fiable result to show for, to sink ones teeth into.

    ashvini, new delhi india.

  6. hi,

    very good explanation indeed. More than your mathematical know how, what really matters is logical approach. But the beauty of this problem is that, the result turns out to be in mathematical form.

  7. @ashvini: Definitely — I need to experience an idea firsthand before I am truly comfortable with it.

    @koushik: Thanks for the comment; yep, limits give us a logical framework to make the best predictions possible.

    @Brit: Awesome, glad you liked it :).

  8. Excellent work as always. One comment.
    “The error margin (epsilon, ε) is the function result, i.e. the position of the ball. “
    I thought the error margin was the difference between the actual position of the ball and the predicted position of the ball.

  9. Whoops, I should clarify, thanks. The error margin is the maximum amount the points in the visible range are allowed to vary from your prediction. Every point in the zoom range must lie within the error margin for us to feel confident.

  10. Thanks a lot! You know, this calculus stuff is not really in my syllabus but i have a big interest in physics and as we all know, it’s close to impossible to appreciate many higher level concepts of physics without a thorough knowledge of calculus and so i decided to go through those heavy books on this haunted topic and guess what, i derived some sort of half baked knowledge but a big thanks to you that i was finally able to understand the basics of calculus and make certain crucial amendments to my foundation.

  11. wonderful explanation !!!!!!even a kid can understand limit through ur article. but i have a question. math has black hole type scenario like infinty ‘somathing divided by zero etc .r irrational numbers also behave like limits because we just approach them not get them.i mean we just
    get closr to them not precisely evaluate them .waiting for an article on limits and irrational number from god of mathmatical explanation (kalid sir)

  12. @Abhineet: Awesome, glad it’s helping. Cementing the foundation for ideas is great.

    @Jar: Glad it helped!

    @Gulrez: Happy it worked. I’m not well versed in number theory, but irrational numbers (like e, pi) can be defined as limits, i.e. the result of some process that continues forever (after all, how many sides do you put on a shape to make it a circle?). I’d like to do more on this.

  13. In your epsilon-delta example, you have epsilon in units of distance (+/- 0.1 meters) and delta in units of time (+/- 0.1 seconds), so the units on one side of the inequality do not balance with the units on the other side. Since I teach physics and not math, this was confusing to me. Could you please explain? Otherwise I find your explanations extremely helpful and I plan to continue this series once I get past this obstacle. I remember working very hard at this in college very many years ago without truly understanding it, but now I’m on the verge of actually understanding it.
    Thanks
    Joe

  14. Hi Joe, great question. Notice we actually have 2 separate inequalities, essentially:

    * If time (the function input) is within a certain range, then distance (the function output) must be within a certain range.
    * i.e., when 0 < |x − c| < δ, we have |f(x) − L| < ε

    Time and distance are never compared directly, just corresponding times and distances. Time of x results in distance of f(x), but x and f(x) never appear in the same inequality. You’re right though, it wouldn’t make sense to compare them, and applying units is a good check to see if the variables have been mixed along the way.

    —-
    Whoops! I re-read what you wrote, and understand better. In the definition of the limit, the two quantities are not compared. But when comparing the conditions that makes each meet its threshold, they could be. Here’s how I see it:

    We are comparing inputs (seconds) and outputs (meters) and trying to equate them, not from a “units” perspective, but from an accuracy one. I.e., how does a “meter” of accuracy translate into seconds? (An accuracy of +/- 1 meter may require a time interval of +/- 0.1 seconds). The “meter” and “second” aren’t really the SI units anymore, they are inputs and outputs in a particular system [because in a different function, a meter of accuracy may require more seconds, or may not be possible at all if the function oscillates wildly].

    In other words: what range of meters has the same accuracy as a given range of seconds? (“Ranges of precision” between the inputs and outputs can be compared, even if the units can’t be.)

  15. Okay. Thanks. I was confusing the variable “x” with position, since I use it so often that way; but in your example “x” is time and f(x) is the position. I worked it through using “t” for time, and I understand it now.

    Joe

  16. Quick clarification/correction: In your Flipping Zero and Infinity section, you have an error. Since \displaystyle{x\to +\infty}, you must have that \displaystyle{\frac{1}{x}\to 0} *from the right*, and thus\displaystyle{y\to 0^+}. Having \displaystyle{x\to\infty} is a one-sided limit, but stating \displaystyle{y\to 0} is a two-sided limit.

    This is a source of many an error on an AP exam…

  17. Hi Tim, that’s an excellent addition, and something that would have tripped me up as well! Appreciate the note, I’ll revise the article.

  18. Hi Kalid. Nice article. You said the number of integers is neither odd nor even , but I guess it is clearly odd . How ? well let the number of positive integers be x . Then number of negatives is also x. There is one more integer remaining , 0 . Thus number of integers is 2x+1 , which is clearly odd. So why do you say that we can’t tell whether number of integers is odd or even ?

  19. Hi Kalid,

    I really enjoyed your gentle introduction to calculus and the finding pi articles. I just recently purchased your book as a token of appreciation.

    I have a question regarding the zoom levels; it was stated that:

    “The predictions agree at increasing zoom levels. Imagine the 3:59-4:01 range was 9.9-10.1 meters, but after zooming into 3:59.999-4:00.001, the range widened to 9-12 meters.”

    I don’t see how zooming in increased the range. If you’re looking more precisely, then wouldn’t the range be much smaller?

    Cheers,

    Dave

  20. @A Googler: Good argument! In my head I was thinking about the number of positive integers, but being able to match them up like that might make the resolution more clear. (I’m still not sure if an infinite number is “allowed” to be even or odd, but if it stays odd as it “grows”, maybe?).

    @Dave: Thanks for the support! Good point, it’s not really possible that the range would *increase* as you zoomed. Imagine the range never diminishing though — things not getting more accurate at all as you zoomed in. Then your confidence/predictions wouldn’t be greater as you looked closer, and you wouldn’t feel comfortable in your predictions. Excellent feedback here!

  21. Hi Kalid what a great website you have, really enjoyed your article!

    Anyway, talking about limits I still have some questions:

    1. Why do we need limits?
    Say that you have an equation which results an indeterminate form 0/0. Then you make a simplification to find the limit. Since limit means “as x approaches to..” then the result is not the exact answer, right? What confuses me is what advantage or benefit that we get by knowing the limit. I mean we know the result cannot be determinated, but we still insist to get its limit? What for? Do you agree that limit is not a certain answer, and if you do (or if it’s true) this will lead us to my 2nd question

    2. If my concept (or mindset) about limit as an uncertain thing is true, then derivatives and integral suppose to be uncertain things. Is it true?

    Please correct my mindset if it’s wrong. These limit, derivatives, and integral things are driving me crazy right now. I really want to understand the analogy, logic, mentality, etc of these matters

    Cheers :)

  22. Thank you so much for taking the time to explain to people about math in ways that people can actually relate to. You are awesome and will go far! You should definitely think about teaching as a professor.

  23. What you do is brillllliant. I’m a student in the eleventh grade, and I think your website is really opening up new avenues- and I’ve been here for exactly 15 minutes! I’m definitely coming here more, and wow. Hats off to you, dude. :D You should be very proud. :’) :D

  24. A bit late, but here goes anyway :)

    @Raifu: I think limits are useful, even for indeterminate forms like sin(x)/x, because we can get a reasonable idea for a starting point. Many situations begin at t=0, but if we have t as the denominator, we technically have an indeterminate form. But we “know” that the position at t=0 was valid, so limits give us a nice estimation of what it should be.

    Derivatives/integrals only work for functions that match their limit at every point, which are called continuous. You’re correct though, there are functions which are ill-behaved (do not match their limits) and we can’t use the regular calculus tools on them! (I’m not super familiar with this, I just know that if a function is discontinuous you need to be very careful, and work around the discontinuity, etc.)

    @Diane, Anonymous: Thanks so much! Glad you’re enjoying the site. I’d like to keep exploring more avenues to explain things :).

  25. Hi Omer,

    I’d say the main difference is that infinitesimals create a new class of numbers (which are too small to measure with our existing numbers), and limits stay within our number system (making a solid prediction about what happens if we *could* have our number disappear).

    Functionally, the results are the same, but I prefer infinitesimals because we can treat dy, dx, etc. as actual microscopic quantities (similar to physics). Technically, with limits, you’re not allowed to separate dy/dx into variables (dy/dx is a shorthand for a larger limit).

  26. Great post as always Kalid!

    I’ve seen a few indications here of the use of limits toward the task of making sense of some indeterminate forms (e.g. 0/0 but there are others). It may only be my unique myopia but this use of limits has always been only in my periphery. I’ve always thought of limits as, well, to be honest, an essentially useless bit of formalism that we feel the need to put in our proofs lest the demagogues of Rigor chide us too harshly. But then I started to appreciate the role of Rigor… At least I have seen enough to try to fight the transformation to demagogue.
    Enough philosophy.

    In my mind there’s always been a more natural method to attack the indeterminate forms in their varied flavors (0/0, x/0, inf/inf) and that is l’Hopitals rule. Lets just pretend that limits don’t actually appear in l’Hopitals rule. I would like to see how you present this topic as I don’t see it in the site guide.

    I see l’Hopitals rule answering the question better because it talks about rates of change (derivatives) instead of the limit talking about how close can you get to ‘there’ without actually getting there.
    Limits are like two cars playing chicken and assuming neither will turn. Then we ask what happens to drivers at the point where they are infinity close to crashing but not quite there. It just seemed to me a bunch of mathematic trickery.
    I say l’Hopitals rule (as I delude myself and ignore the limits in the definition) more like two lemmings running toward a cliff with a trough tied between them and a marble in the trough. We then ask, which way will the marble roll as the lemmings approach oblivion? I don’t have to look at really, really close but not quite there. I just ask which lemming runs to its death faster? If the lemming on the left has a higher rate of change then I can predict the marble will tilt left as the trough goes over the edge. If I have f(x)/g(x) and I know they both reach 0, I just ask which rushes toward 0 faster, rate of change of f(x) with respect to x or df/dx.

    I’ve seen a few receptions of the idea that infinity is not a number, just a concept. I’ve heard the same from every instructor who writes it in an equation and insists it has all the properties of a number . I throw in this next little tie bit because I can’t resist. Its a question I often pose to myself in an effort to gain more insight, though I usually just puzzle myself further:
    If I say ‘You never reach infinity’, is that not equivalent to saying ‘You reach infinity at never’?

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