I've been delving into the concept of limits and the Epsilon-Delta definition. The most basic definition, as I understand it, states that for every real number $\epsilon \gt 0$, there exists a real number $\delta \gt 0$ such that if $0 \lt |x - a| \lt \delta$ then $|f(x) - L| \lt \epsilon$, where $a$ is the limit point and $L$ is the limit of the function $f$ at $a$.
While I grasp the formal definition, I'm grappling with the philosophical aspect of it. Specifically, I'm questioning whether this definition truly encapsulates our intuitive understanding of what a limit is. The idea of a limit, as I see it, is about a function's behavior as it approaches a certain point. However, the Epsilon-Delta definition seems to be more about the precision of the approximation rather than the behavior of the function.
In the book "The Philosophy of Mathematics Today" by Matthias Schirn,on page 159, it is stated that: "At one point, Etchemendy asks: 'Howdo we know that our semantic definition of consequence isextensionally correct?' He goes on to say: 'That [this question] nowstrikes us odd just indicates how deeply ingrained is our assumptionthat the standard semantic definition captures, or comes close tocapturing, the genuine notion of consequence' (Etchemendy 1990, 4-5).I do not think that this diagnosis is correct for some people: forsome logicians, the question is similar to: How do we know that ourepsilon-delta definition of continuity is correct?".
This quote resonates with my current dilemma. Does the Epsilon-Delta definition truly capture the essence of what we mean by a 'limit'? though the epsilon-delta definition is a mathematical construct, what evidence do we have that it accurately reflects our intuitive concept of a limit? How can we be sure it is not merely a useful formalism, but a true representation of the limit as a variable approaching some value? Are there alternative definitions or perspectives that might align more closely with our intuitive understanding of limits? I would appreciate any insights or resources that could help me reconcile these aspects of the concept of limits.Thank you in advance for your help.
edit:i think i should add my motivation of asking the question, what i really want is an argument which can demonstrate that this definition of limit is the definition of limit which no better definition can come up, i can accept the definition as it is in its own axiomatic system and in itself, but whats the certainty that a hundred years from now we come up with a better definition still? its not about the thing that we cant understand i am more worried it there is something out our sphere of recognition if we are not taking note of, because everybody just seem to accept the definition without any further doubt an examination.