# A rigorous definition of a limit and advanced calculus

## Main Question or Discussion Point

i'm trying to review calculus and look a little deeper into proofs/derivations/etc. i'm doing this both for fun and to review before i go back to school.

am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page: http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

i think i can get this if i spend some more time with the explanations and exercises on this page. my question is, are there lots of hard to understand pieces of logic like this in advanced calculus or analysis? i don't have to take analysis for my degree but i want to learn a little bit of it just to expand my understanding of how the universe works (which is fun).

when i took calculus i pretty much worked through the textbook and i remember having difficulty with the definition of the limit in that book as well. for those who have taken analysis, i guess, what is it like? what can you say about it to someone with beginner/intermediate level experience like myself? i remember reading that Andrew Wile's proof of Fermat's Last Theorem was over 100 pages long. is there anything in advanced calculus like that, or are the main theorems relatively easy to demonstrate?

also, any tips on how to understand this epsilon delta thing would be appreciated.

edit: maybe the super scientists of the future will be AIs? apparently there are some super long proofs that were done by computers - not really AIs, but sophisticated programs. this got me thinking so i did some google searches.

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jbriggs444
Homework Helper
2019 Award
The way I like to think of the epsilon delta stuff is as a kind of two player game. We are given a function f() and an x coordinate.

a. I pick a number y, propose it as the limit of f() at x and play begins.
b. The other player picks an epsilon -- an non-zero error tolerance in the y direction.
c. My job is then to pick a delta -- a non-zero distance in the x direction.
d. Finally, the other player picks an x' within delta of x (but not equal to x).

If f(x') is more than epsilon different from y then the other player wins.
If f(x') is less than epsilon different from y then I win.

The limit is defined and equal to y if I can come up with a strategy that always wins.

fresh_42
Mentor
i'm trying to review calculus and look a little deeper into proofs/derivations/etc. i'm doing this both for fun and to review before i go back to school.

am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page: http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx
This is the definition of what it means for a function to be continuous at a point $x=a$. It is a bit tricky, because it is very formal. It might help to imagine, which functions are not continuous at $a$.

The definition has to rule out these situations.
i think i can get this if i spend some more time with the explanations and exercises on this page. my question is, are there lots of hard to understand pieces of logic like this in advanced calculus or analysis?
This depends on what you want to learn and what you consider advanced. There is a long way to the top.
i don't have to take analysis for my degree but i want to learn a little bit of it just to expand my understanding of how the universe works (which is fun).
As so much in life: all a question of practice.
when i took calculus i pretty much worked through the textbook and i remember having difficulty with the definition of the limit in that book as well. for those who have taken analysis, i guess, what is it like? what can you say about it to someone with beginner/intermediate level experience like myself?
You can always come to PF and ask. Usually, in I think in your case always, there will be someone who can help you. I've had similar problems as I first saw this definition. And still today I prefer to look up the order of the "for alls" and "exists" if I need them.
i remember reading that Andrew Wile's proof of Fermat's Last Theorem was over 100 pages long. is there anything in advanced calculus like that, or are the main theorems relatively easy to demonstrate?
As far as I know is his proof mainly algebra and I think the theory of modular forms. Trying to understand it is in itself a lifetime goal. Most mathematicians don't.
also, any tips on how to understand this epsilon delta thing would be appreciated.
Play a little with the definition and try to understand, why the functions in my image fail to be continuous at $x=a$ and why they are continuous elsewhere: step, gap, asymptote.

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thanks guys, both of those were helpful. if today weren't game night for me, with me on my laptop as my friends and i were watching the Dark Tower and a movie about bigfoot, i wouldn't have had the patience and persistence to sit and study this. i was good at solving calculus problems, but this type of thing is another layer of theory and difficulty. i don't quite "grok" it yet, but i think as i do some examples and think about it i will become comfortable enough with it. attention span is important for this kind of thing!

i thought i could just understand theory and not do problems, but i guess you've got to do a bit of both sometimes if you want to wrap your mind around a concept.

Stephen Tashi
what can you say about it to someone with beginner/intermediate level experience like myself?
Before you worry about the particulars of the definition of limit, study a little logic, especially the logic of the quantifiers "for each" and "there exists". After you understand how to deal with logic and quantifiers, then return to the particular example of the definition of a limit. There is much material on the web about elementary logic and the logic of quantifiers - e.g. https://www.whitman.edu/mathematics/higher_math_online/section01.02.html

Understanding the logic of quantified statements is essential for studying calculus from a rigorous point of view.

Theres a nice little cheap book that has a great explanation of logic quantifiers. Levin: "Discrete Mathematics." It can be found for free, as it is an open stacks book, or purchased for less than 20 dollars. I suggest the second edition, since the logic is put in the beginning sections and not towards the middle. Also, Hammock: Book of Proof. It has great explanations for the fundamentals of mathematics. i.e. relations, functions, etc.

FactChecker
Gold Member
The "epsilon-delta" definition of a limit is used everywhere in analytical mathematics. If you want to be good at analytical mathematics, you will want to become very comfortable with this type of definition and with proofs using it. The definition may seem obscure at first, but learning it will pay off and you will use it so much that it will become second nature.
In english:
No matter how close ($\forall \epsilon > 0$) you want to keep a function to it's limit, $L$, ($|f(x)-L| < \epsilon$) for x near $a$,
You can find a $\delta$ ($\exists \delta \text{ such that}$), where keeping x that close to $a$ ($|x-a| < \delta$) will keep f(x) close enough to $L$. ($|f(x)-L| < \epsilon$)

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Aufbauwerk 2045
I will add my two cents worth. I had this math teacher at U. once who boasted about how he didn't care about "delta-epsilonics" since he was too busy working on practical stuff like matrix calculations for jet fighter aerodynamics or whatever. Anyway, please don't listen to guys like that particular math teacher of mine, who build up the definition of a limit using delta and epsilon into some kind of big obstacle, or something for the pedantic types which you really don't need to worry about. I really hate it when teachers shortchange their students like that guy did.

Actually if you want to really learn calculus, you should spend a lot of time on limits and learn how to prove a limit using delta-epsilon. It takes some practice to get the idea.

I recommend Bob Miller's math books in general because he usually explains things very clearly. You will find a very good introduction to limits in chapter 1 of his Calculus for the Clueless Part 1. I'm not familiar with any better introduction.

The "rigorous definition" is the one you need to know. Who wants to learn non-rigorous calculus? It makes no sense. Miller gently leads you from the basic idea to the hard core rigorous version. After Miller, I would look for more problems in limits from other books and work them all until you are very skilled in the topic. You will be very glad you did.

P.S. since I'm on the topic, I recommend to everyone who wants to learn math to sit down with a book, paper and pencil, and a calculator, read very slowly, pay attention to every detail, and work tons of problems. Old fashioned advice I know, but it works wonders. You don't really understand it until you have worked lots of problems and burned the topic into your long term memory.

Miller has a good story on one topic he wanted to make sure his students really learned. He said if he woke them up in the middle of the night and asked them a certain math question, they would be able to answer it without hesitation, and then tell him to go away so they could get back to sleep.

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Aufbauwerk 2045
i'm trying to review calculus and look a little deeper into proofs/derivations/etc. i'm doing this both for fun and to review before i go back to school.

am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page: http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

i think i can get this if i spend some more time with the explanations and exercises on this page. my question is, are there lots of hard to understand pieces of logic like this in advanced calculus or analysis? i don't have to take analysis for my degree but i want to learn a little bit of it just to expand my understanding of how the universe works (which is fun).

when i took calculus i pretty much worked through the textbook and i remember having difficulty with the definition of the limit in that book as well. for those who have taken analysis, i guess, what is it like? what can you say about it to someone with beginner/intermediate level experience like myself? i remember reading that Andrew Wile's proof of Fermat's Last Theorem was over 100 pages long. is there anything in advanced calculus like that, or are the main theorems relatively easy to demonstrate?

also, any tips on how to understand this epsilon delta thing would be appreciated.

edit: maybe the super scientists of the future will be AIs? apparently there are some super long proofs that were done by computers - not really AIs, but sophisticated programs. this got me thinking so i did some google searches.

A few more thoughts on your question.

No, you are not the only one who has difficulty understanding the rigorous definition of the limit. It took even great mathematicians a while to figure it out. We are lucky, because it's already been figured out for us. As I stated earlier, it's not so hard if you follow Bob Miller's explanation step by step. I would write out all of this examples and go over them until you start dreaming about it.

I think the hardest part of learning calculus is understanding the basis which is limits and continuity. One you understand limits, you can understand the definition of a derivative quite easily. But you still need to apply limits of all sorts to make progress. Not everything is as simple as differentiating a polynomial. You can also understand the definition of the integral. Then you can understand the relationship between the derivative and the integral.

Once you get that down in one variable, I think it's relatively easy to extend to two and three variables. Then it's mainly a matter of doing lots of problems and seeing how calculus is applied. Remember, it was developed because the mathematics was needed to solve real world problems, not just as some kind of interesting puzzle. When you see what kind of problems can be solved with calculus, then you see how awesome it is.

It gets even much greater when you learn about advanced topics. I will single out Green's Theorem. Speaking of genius -- an overused word to be sure -- he was one of the top geniuses. Especially when you consider he only had one year of formal schooling. There is genius, then there is genius. Green was one of the latter.

Even the symbols used in calculus are beautiful. Continue to the stage where you learn about partial differential equations and you have the symbols of the universe at your fingertips. It's the greatest. Beauty, power, understanding -- what more motivation does one need?

To sum up, I would spend two or three weeks doing nothing but limits and also to review anything from pre-calculus you are not sure about, such as logs and exponentials, trig functions, and series. Series are super important in so many ways, not just in calculus.

To make calculus seem more approachable, just remember that it is based on two problems. Problem #1 : find the slope of a line tangent to a curve at a given point. Problem #2: find the area under a curve between two given points. Differential calculus answers problem #1. Integral calculus answers problem #2. The two problems are in fact related, as you learn in first year calculus. These two problems needed to be solved in order for physics to make progress during the time of Newton and Leibniz.

Finally, for now, here is a mystery question for science fiction movie fans. In which movie does a scientist say to some of the other characters, "I have something you don't have -- Calculus!"

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