An Intuitive Introduction To Limits

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:

This is a companion discussion topic for the original entry at

@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 :).

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.

What program you use for illustrations? Thanks

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 ?

Excellent, finally! Well done Kalid. I remember my HS teacher butchering this


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.

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)

Tremendous explanation.the epsilon delta concept is fascinating.thanks

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.

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.

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,


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.

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.

@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.

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.

Hi Majo, I use PowerPoint to make the diagrams.

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.

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.)

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.