@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.
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.
Hi Kalid what a great website you have, really enjoyed your article!
Anyway, talking about limits I still have some questions:
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
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 
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.
Quick clarification/correction: In your Flipping Zero and Infinity section, you have an error. Since $$x\to +\infty$$, you must have that $$\frac{1}{x}\to 0$$ from the right, and thus$$y\to 0^+$$. Having $$x\to\infty$$ is a one-sided limit, but stating $$y\to 0$$ is a two-sided limit.
This is a source of many an error on an AP exam…
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
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. 
You should be very proud. :’)
@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!
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 :).
Kalid,
In s nutshell, what is different between the limit approach and the infinitsimal approach?
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).
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’?
Hi Khalid–
I like infinitesimals too. Even if they’re not traditional numbers, they don’t bother me at all. To me they’re just like infinite sums–like an infinite series that converges on a number or a point. And that’s just what an integral is–an infinite sum, correct? So even though infinity is inherent in the notion of an infinite sum, it is acceptable to mathematicians because it doesn’t go shooting off into the unknowable.
In any case, I have a related question: Would it make you or the math community squeamish to consider a limit as an example of what might be called “finite infinity”? A “finite infinity,” in my view, is simply a converging series, an infinity that has a finite limit.
I ask because because Emily Dickinson uses this very phrase, and I don’t think she was taking poetic license. Though not well-versed in mathematics, she had an intuitive understanding of the subject. She knew, for example, that there is no difference between infinity and infinity minus one. And she knew as well as Cantor did that infinity comes in different sizes.
So what do you think? Is “finite infinity,” in your opinion, an acceptable mathematical term? Could I use it without embarrassing myself in a forum that might be read by math as well as poetry geeks?
Thanks.
–Tim
Hi Kalid,
I just found your site, and it’s wonderful!
Regarding the intuition of limit: after years of struggling with it, I came up with an insight that finally satisfied me. I would be very happy to hear your opinion.
The epsilon/delta definition of limit formalizes the concept of gapless extension (or unbroken continuation): Let f be a function with a “black hole” at point c (using your nice analogy). We wish to extend its definition to include f© as well. Suppose we have a candidate value L for f©. Now, if we can show there are no gaps between L and the neighboring values of c, then L is the “most consistent” candidate and we can happily define f© = L
This gaplessness is shown with the formal definition of limit:
For any ε > 0 (i.e. for any potential gap between L and the neighboring values of the black hole) there is a δ > 0 (a neighborhood of c) such that for all 0 < |x − c| < δ we have |f(x) − L| 0, it follows that there are no gaps whatsoever between L and the neighboring values. So L is the perfect candidate for f© = L. Similar arguments can be presented for limits of successions, series, integrals etc…
Now, why is gaplessness the best criterion to extend a set of known values to an unknown value? Because nature works that way, at least macroscopically. A basic principle of natural philosophy is “Natura non facit saltus” (Nature does not proceed by jumps, http://en.wikipedia.org/wiki/Natura_non_facit_saltus). Therefore by defining e.g. instantaneous rate of change as the gapless extension of Δy/Δx values to the case Δx = 0, we are working according to the (macroscopic) laws of nature, which would account for the incredible success of calculus.
I would love to hear your opinion on the above. Again, thanks for your wonderful and enlightening site.
– Max
Hi Kalid,
Sorry for posting twice, but the previous comment got somehow messed up…
I just found your site, and it’s wonderful!
Regarding the intuition of limit: after years of struggling with it, I came up with an insight that finally satisfied me. I would be very happy to hear your opinion.
The epsilon/delta definition of limit formalizes the concept of gapless extension (or unbroken continuation): Let f be a function with a “black hole” at point c (using your nice analogy). We wish to extend its definition to include f© as well. Suppose we have a candidate value L for f©. Now, if we can show there are no gaps between L and the neighboring values of c, then L is the “most consistent” candidate and we can happily define f© = L.
This gaplessness is shown by the formal definition of limit:
For any ε > 0 (i.e. for any potential gap between L and the neighboring values of the black hole) there is a δ > 0 (a neighborhood of c) such that for all 0 < |x − c| < δ we have |f(x) − L| 0, it follows that there are no gaps whatsoever between L and the neighboring values. So L is the perfect candidate for f© = L. Similar arguments can be presented for limits of successions, series, integrals etc…
Now, why is gaplessness the best criterion to extend a set of known values to an unknown value? Because nature works that way, at least macroscopically. A basic principle of natural philosophy is “Natura non facit saltus” (Nature does not proceed by jumps, http://en.wikipedia.org/wiki/Natura_non_facit_saltus). Therefore by defining e.g. instantaneous rate of change as the gapless extension of Δy/Δx values to the case Δx = 0, we are working according to the (macroscopic) laws of nature, which would account for the incredible success of calculus.
I would love to hear your opinion on the above. Again, thanks for your wonderful and enlightening site.
– Max
Hi Max, glad you’re enjoying the site!
Wow, that’s a great way to think about it. When a limit isn’t defined at a point (we have a black hole), it’s almost like a literal tearing/hole in the fabric of the function :). Figuring out what F© = L would “repair” the hole, i.e. make a seamless / gapless transition between values. I do like this way of thinking about it! (You can’t observe F©, but what value L would perfectly “mend the gap”?).
I havent posted in a while and had to comment.
I like your thought process, Max, it makes me think. I would offer a caution about “Nature does not progress in jumps”. Even if we don’t like the idea, quantum physics demonstrates clearly that nature does in all cases progress in jumps. Some even theorize that it could not possibly be otherwise.
In so far as I pose that caution I also say, press forward. Think for yourself and challenge the “status quo erat”, that is how we advance. It is in asking questions like yours that I have begun to gain a small insight into the idea of a continuum and different levels of a continuum (search this site for a description of aleph naught).
Excelsior,
Eric
Praising article…found it a lot helpful! Thank you very much my friend.
Adorable,Kalid!I don’t understand the ball position in soccer example that you are referring to,from where you measure ball’s position?could you let me know?
Love the site and your posts. Pardon my ignorance, but… “aleph me alone” = ???