Proving the formal definition of límit through the intuitive definition of límit written formally

[MxIgoResEs]

Trying to understand the definition of limit I realized that none of the explanations given are really convincing because writing the intuitive definition of limit formally:

It is impossible to realize immediately that it is equivalent to the definition of limit:

I have read a bit of first-order logic to be able to demonstrate with its laws and mathematical laws that the 2 definitions are equivalent, but I did not succeed. Does anyone know first order logic and can prove that the definitions are equivalent? would solve this dilemma for many students.

 

[Otis]

Hi MxIgoResEs. I'm not sure what the diagram caption means by "cannot get as close as desired", but, really, with a discontinuity like the one shown, we ought not say, "as x gets closer to 3, f(x) gets closer to 2" because that statement is not correct when x approaches 3 from the right.

\(\displaystyle \lim_{x\to 3^{+}} f(x) = 3\)

\(\displaystyle \lim_{x\to 3^{-}} f(x) = 2\)



[imath]\;[/imath]

 

[JeffM]

The two statements are not equivalent so it is no wonder that you cannot prove them to be rquivalent.

One is a general definition of limit. The other is a definition of limit that applies in a special case. Notice that the second statement is incorporated as just part of the first statement.

 

[MxIgoResEs]

The two statements are not equivalent so it is no wonder that you cannot prove them to be rquivalent.

One is a general definition of limit. The other is a definition of limit that applies in a special case. Notice that the second statement is incorporated as just part of the first statement.

Sorry if I didn't write the intuitive definition of right limit formally correctly (as x gets closer to a from the right, f(x) gets closer to L and you can get as close to L as you want, just like in the books). Can you give it correctly and show that it is equivalent to the 2d proposition?.
I use the right-hand definition of limit instead of the full (both-sided) definition of limit because it is simpler to write in mathematical notation

 

[JeffM]

I do not even know what you mean about an intuitive definition being written in a way that is formally correct. A definition is either formally correct or not, whether or not it is intuitive.

Your first definition applies only over intervals where f(x) is monotonically increasing or decreasing. Your second definition applies generally to any function. They are not talking about the same thing. More importantly, your first definition includes the second definition with the sole difference being that in the first there is an implied definition of

[math]L = \lim_{x \rightarrow a^+} f(x) \text { if } \lim_{x \rightarrow a^+} f(x) \ \exists.[/math]
It is a trivial difference in notation

 

[MxIgoResEs]

I do not even know what you mean about an intuitive definition being written in a way that is formally correct. A definition is either formally correct or not, whether or not it is intuitive.

Your first definition applies only over intervals where f(x) is monotonically increasing or decreasing. Your second definition applies generally to any function. They are not talking about the same thing. More importantly, your first definition includes the second definition with the sole difference being that in the first there is an implied definition of

[math]L = \lim_{x \rightarrow a^+} f(x) \text { if } \lim_{x \rightarrow a^+} f(x) \ \exists.[/math]
It is a trivial difference in notation

I mean writing the intuitive definition in mathematical and logical symbols. The part of the intuitive definition "As x gets closer to a, f(x) gets closer to L" is only true if the function is nonincreasing or nondecreasing by so this statement should be written as part of the intuitive definition. For example the sine function, the limit as x approaches 0 is 0 but the values of f(x) approach and move away from 0 if x approaches a taking values from say 2 pi. That it be monotonically increasing or decreasing in (a,b) is a condition that must be eliminated, the problem is how.

 

[JeffM]

But it is not true that a limit exists only if a function is non+decreasing or non-increasing. I am not exactly sure where your confusion is arising.

Consider the function [imath]\dfrac{sin(x)}{x}[/imath] What is the limit as x approaches 0? It is 1, but the function itself is not even defined at x = 0 so it is neither consistently non-increasing nor consistently non-decreasing in any interval containing 0.

Here is where I suspect your confusion lies. To show that a right limit of f(x) exists at c, we must define an interval such that

[math]L = \lim_{x \rightarrow c} f(x) \ \exists \iff \exists \ a \text { such that } c < x < a \implies |L - f(x)| < \epsilon \text { for arbitrary } \epsilon > 0.[/math]
That is a definition. We do not require f(x) to be non-increasing or non-decreasing. What I suspect you are thinking is this proposition: if a right limit exists at c, there is an interval (c, d) where the epsilon test is met AND the function is either monotonically increasing or decreasing for any positive epsilon. But that is irrelevant to the definition of the right limit. We do not need to find d; we only need to find a. It is necessary to find a. It is sufficient to find d.

 

[MxIgoResEs]

But it is not true that a limit exists only if a function is non+decreasing or non-increasing. I am not exactly sure where your confusion is arising.

Consider the function [imath]\dfrac{sin(x)}{x}[/imath] What is the limit as x approaches 0? It is 1, but the function itself is not even defined at x = 0 so it is neither consistently non-increasing nor consistently non-decreasing in any interval containing 0.

Here is where I suspect your confusion lies. To show that a right limit of f(x) exists at c, we must define an interval such that

[math]L = \lim_{x \rightarrow c} f(x) \ \exists \iff \exists \ a \text { such that } c < x < a \implies |L - f(x)| < \epsilon \text { for arbitrary } \epsilon > 0.[/math]
That is a definition. We do not require f(x) to be non-increasing or non-decreasing. What I suspect you are thinking is this proposition: if a right limit exists at c, there is an interval (c, d) where the epsilon test is met AND the function is either monotonically increasing or decreasing for any positive epsilon. But that is irrelevant to the definition of the right limit. We do not need to find d; we only need to find a. It is necessary to find a. It is sufficient to find d.

I wrote f(x) MUST be non-increasing or non-decreasing because it is the most intuitive form of limit (as x gets closer to a, f(x) gets closer to L) although it is not the only way to write the intuitive definition. For example I found intuitively an alternative to "f(x) nondecreasing or nondecreasing on (a,b) and ∀x1, x2∈(a,b), x2<x1⇒|f(x2) - L|<| f(x1) - L|" just "∀x1∈(a,b),∃x2∈(a,b): x2<x1 and f(x1)≠L⇒|f(x2) - L|<|f(x1) - L|" (this means that there is always a value closer to a for x so that its f(x) is closer than L, must f(x1)≠L because if f(x) is not defined at a, there would be no x2 that fulfill the condition). I think that starting from another definition that is still intuitive is better because from what they have answered me and it shows that it is somewhat difficult to prove it.

 

[JeffM]

I have never liked standard analysis. But I do get that it is logically consistent and (I believe) minimalist in its axioms. I do not see that your non-increasing or non-decreasing condition is necessary to define left or right limits, and the condition obviously does not apply to limits generally. The primary use of right and left is to prove that a limit exists.

But I think we are now talking about matters of taste, and am not going to argue about that. You initially asked how you prove that two definitions are logically equivalent. I do not think you can. One has stronger constraints than the other.