A question about the formal definition of limit

[flamengo]

Is it possible to learn to prove limits by the formal definition without doing a course of real analysis? I'm not talking about just following the model that the Calculus books give, what I want is to understand the why of all the steps in formally proving the limit, to understand the why to use inequalities that way. Anyway, is this possible without studying Real Analysis (only with a course of Calculus)?

 

[fresh_42]

Is it possible to learn ...

Yes. It is always possible.

...to prove limits by the formal definition without doing a course of real analysis?

If you're learning the concepts, it is automatically a kind of course. You may not need to formally attend a course, but if you do it all alone, you most probably need more time and you won't have systemic corrections if you go wrong. Therefore it will take longer to get false understandings out of your mind again.

I'm not talking about just following the model that the Calculus books give, what I want is to understand the why of all the steps in formally proving the limit, to understand the why to use inequalities that way.

An example of what you mean and what not would be helpful. I have difficulties to see one apart from the other. You could start to tell whether you are talking of limits of sequences, series (which is basically the same) or approaching function values as it is the case in the different concepts of continuity (pointwise, uniform, Lipschitz) and differentiability. There is even a pure topological approach possible.

Anyway, is this possible without studying Real Analysis (only with a course of Calculus)?

Yes. In the end it is only a matter of time, effort and diligence. But whatever you do, I would try to learn in a team or at least get some help here on PF in order to check whether your understanding is on the right track - some kind of correction mechanism.

 

[fresh_42]

Here is a list of insight articles which might be helpful for a more detailed answer.
(More are available for linear algebra, geometry and so on. You can find them via the search routine in our Insights section under the keyword "self study".)

https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/
https://www.physicsforums.com/insights/self-study-calculus/
https://www.physicsforums.com/insights/problems-self-studying/
https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/
https://www.physicsforums.com/insights/self-study-analysis-part-ii-intermediate-analysis/
https://www.physicsforums.com/insights/how-to-study-mathematics/
https://www.physicsforums.com/insights/overcame-learning-challenges-faced-studying-stem/

 

[Stephen Tashi]

Is it possible to learn to prove limits by the formal definition without doing a course of real analysis? I'm not talking about just following the model that the Calculus books give, what I want is to understand the why of all the steps in formally proving the limit, to understand the why to use inequalities that way.

You won't understand limits (and many other advanced mathematical concepts) unless you understand the logic of quantifiers ("for each", "there exists"). To understand the logic of quantifiers, you can read a textbook on logic. Of course, some people learn the logic of quantifiers by banging their heads against a lot of mathematics, but it's simpler to read a logic book.

After you understand quantifiers, then you can study the special tricks with inequalities that are used in proving results about limits. There is no specific procedure or algorithm that works for all proofs involving limits.

If we distinguish between "real analysis" and "calculus", you can learn to do proofs about limits by studying the proofs given in a calculus book that's written in a rigorous style. (Most students of calculus don't study the material in that depth.) Calculus books don't give a specific model for proving results about limits, do they? Students are often asked to prove results about limits of linear functions and that type of problem does follow a predictable pattern.