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I’ve made several posts about this topic, but maybe I just over complicated the epsilon-delta definition. I’d like to ask for help with this. I am completely understand the limit I just not sure the link between the epsilon-delta and intuitive definition that is why I am asking you to assure me or refute me. Is this the link between the intuitive definition and the epsilon-delta definition what I think?

So I think that if my thinking is correct, then there are two possible interpretations of the epsilon-delta definition:

  1. If I understand the epsilon-delta definition correctly, one way to interpret it is that the conditions 0<∣x−c∣<δ and ∣f(x)−L∣<ε essentially describe distances, since absolute value represents a distance or a metric. d=∣a−b∣ means the distance between points a and b. So when I write ∣a−b∣ < r where r>0, it means the distance between a and b can be made arbitrarily small.This is an intuitive way to approach the concept of a limit, and it’s similar to the Heine definition, where we actually move x toward c step by step, and f(x) approaches the limit point L on the y-axis. intitive

  2. The second interpretation is to treat these conditions as mathematical neighborhoods, i.e., intervals that can be made arbitrarily small. We place an epsilon interval around L on the y-axis and a delta interval around c on the x-axis. The idea is: if for every epsilon we can find a corresponding delta such that every x within delta of c (excluding c itself) leads to an f(x) that lies within epsilon of L, then the limit of f(x) as x→c is L. This is the usual way the epsilon-delta definition is taught, not the first one — that one is more like the Heine definition, which captures the intuitive aspect. Usually, people illustrate this with small boxes to “trap” the function value near L as x gets closer to c. interval 1. interval 2. interval 3.

So, is this the correct way to understand the epsilon-delta definition?

Bonus question: Does interpretation (2) somehow imply interpretation (1)? Or am I just over complicating things? Is there any better way to interpret epsilon-delta and see the second interpretation is the same as the first interpretation (the intuitive limit definition)?

Thank you!

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To me it looks as if interpretations (1) and (2) both say the same thing.

Teaching this to newbies, I emphasize two things:

(1) That a function keeps getting closer to a particular number does not mean that that number is the limit. For example: $$ 5.1, \quad 5.01, \quad 5.001, \quad5.0001, \quad \ldots $$ This sequence keeps getting closer to $4.$ And to $3$ and to $2.$ But the limit is not $4$ or $3$ or $2.$

(2) That a particular number is the limit does not mean the function keeps getting closer to that number. For example: $$ \begin{array}{lclcccc} 5.1 & \longrightarrow & 5.01 \\ & \swarrow \\ 5.\underbrace{0\ldots0}_\text{10 “0”s}1 & \longrightarrow & 5.001 \\ & \swarrow \\ 5.\underbrace{0\ldots\ldots0}_\text{100 “0”s}1 & \longrightarrow & 5.0001 \\ & \swarrow \\ 5.\underbrace{0\ldots\ldots0}_\text{1000 “0”s}1 & \longrightarrow & 5.00001 \\ & \swarrow \end{array} $$ Going downward along either the left column or the right, one keeps getting closer to $5,$ but the sequence indicated by the arrows does not keep getting closer to $5;$ it alternately gets closer and farther away. But its limit is $5.$

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When you write that "the conditions 0<∣x−c∣<δ and ∣f(x)−L∣<ε essentially describe distances, since absolute value represents a distance or a metric" this is basically true. However, does a phrase stays at the level of an intuitive definition of the limt which already appeared in luminaries like d'Alembert, Lacroix, and Cauchy. Such a definition falls short of the precision expected in modern mathematics (clarified in the last third of the 19th century decades after Cauchy died), because it does not spell out the order of the quantifiers and the dependence of $\delta$ on $\epsilon$.

So I would say that your intuitive definition 1 is equivalent to what was already practiced around 1800, whereas your definition 2 is a precise formalisation of definition 1.

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