One sentence summary: I think what I am asking about is "how" this exact, particular (and not some other) definition got formulated, came to be what it is, the concepts, background, context that were there in the history of math before it that led to it. Maybe this article answers this question to some extent.
I am having a problem in understanding the $\epsilon$ - $\delta$ definition of limit of a function. Actually, I understand what the definition says perfectly, that, no matter how close we get to $L$ if we can find an interval around $c$ $(x\rightarrow c)$ for which every $f(x)$ is in that interval around $L$, then $L$ is called the limit as $x$ $\rightarrow$ $c$.
The problem is, I'd say, I don't have any solid feel or comprehensive understanding for what the definition is really trying to say. I avoided saying "intuitive understanding" because then people comment like "Everyone's intuition is different", "Forget intuition, just consider the definition", etc. I'm just not having the total clarity that one has after studying any mathematical concept.
I haven't studied anything about sequences or the limit of a sequence concept, which appears before the limit of a function definition in many famous real analysis textbooks. I wonder if this is the reason behind the fuzziness.
So, it would help if you clarify. If you've two cents to directly attack the question of better understanding the $\epsilon$ - $\delta$ definition, please go ahead.
Elaboration after editing: This definition seems to be describing a particular behavior that a function (or the $y$-coordinate) may or may not exhibit around a point $L$ (the limit) when $x$-coordinate comes closer to a point $c$. This behavior is the definition: no matter how close we get to $L$, we can find an interval around $c$ $(x\rightarrow c)$ for which every $f(x)$ is in that interval around $L$. My questions are:
- What are other possible behaviors?
- Are these other possible behaviors described anywhere in Math? If they are, then it provides a broader picture to think about, in which our definition of limit fits in as one of the behaviors that a function can exhibit and one that we are especially interested in for all the applications it has.
So I want to understand the context, background, the bigger picture that this fits into. I want a better way to make sense of it.
Again, I don't know if not having learned the concept of the limit of a sequence has caused this problem.
Some more elaboration: Suppose Cauchy comes in front of me and suddenly out of nowhere gives his definition. I would ask, "Why? Why not define this concept some other way?" For example, while thinking, I constructed the following definition very carefully:
- $\exists f(x) \in \mathbb{R}, \forall\epsilon>0 $ around $L$
- We consider two cases, in some interval $\delta >0$ around $c$, from left and from right-- As $x \rightarrow c$ from left: $\forall x_2>x_1$, $|f(x_2)-L|<|f(x_1)-L|$; As $x \rightarrow c$ from right: $\forall x_2<x_1$, $|f(x_2)-L|<|f(x_1)-L|$
I say, if the above two conditions are satisfied for a number $L$, then we call it the limit as $x \rightarrow c$.
Now, if your work out, you'll find that my definition works for many many functions. But, for a function like $x sin(1/x)$, it says no limit as $x \rightarrow 0$ because condition (2) isn't satisfied, when in fact there's a limit (by Cauchy's definition), equal to zero.
This is a fundamentally different question. Answers to linked questions describe a "game approach" to understand the definition, and I totally get it, but that's not what I am asking about. I understand the definition in and out, and I have made what I am asking about completely clear in my edits.
