1

I am teaching some high school students on the definition of the limit of a function and having a hard time explaining this to them. In our country's textbook, it is defined by using Heine's definition, that is:

Let $K$ be an interval containing $x_0$ and $f(x)$ be a function defined on $K\setminus \{x_0\}$.

A function $f(x)$ has a limit of a real number $L$ when $x$ approaches $x_0$, if for every sequence $\{x_n\}$ such that $x_n\in K\setminus \{x_0\}$ and $x_n$ converges to $x_0$, the sequence $f(x_n)$ converges to $f(x_0)$.

My problem is that I cannot relate this definition to the previous convergence in sequence using $\epsilon-\delta$ terminology, which was "softly" defined by words, and quite intuitive. (i.e, the difference can be smaller than any given small values. )

A sequence $\{x_n\}$ has a limit of $0$ when $n$ approaches $+\infty$, if $|x_n|$ can be less then any arbitrary small number, starting from some term of the sequence onwards.

A sequence $\{x_n\}$ has a limit of a real number $a$ when $n$ approaches $+\infty$, if $\lim_{n\to+\infty}(x_n-a)=0$.

Is there any way I could build a more natural interpretation of this to high school students, i.e how to translate from sequence to function.

Community
  • 1
Lukest_Dang
  • 11
  • In the first place one should not use the above clause as definition of function limit. In order to prove that $\lim_{x\to3}x^2=9$you have to test more sequences $x_n\to3$ than there are atoms in the universe. – Christian Blatter May 15 '20 at 08:07
  • The sequence-based definition is in fact the more intuitive one. If you don't believe me, try doing this and this. Asymptotic expansions tie in nicely with the sequence-based definition, because they give precise bounds on how fast the sequences converge. – user21820 May 15 '20 at 08:28
  • 1
    @ChristianBlatter: That is incorrect; nowhere in the definition does it say that you must test all the sequences. It is perfectly possible to logically reason about such sequences, and hence use this definition in a very efficient way to prove limits! – user21820 May 15 '20 at 08:43

1 Answers1

1

If you are allowed to use the limit to formulate the limit of a sequence, I guess you could also use it instead of the $\epsilon$ formulation.

As for the natural interpretation, try Vsauce on Supertasks. Although it might not be good as a material for students (since Michael jumps from one thing to another as he does), I think that you might find some scenarios there that should be puzzling at first sight, but can be used to showcase the meaning of a limit.

Once you have that solved try to apply it to the limit of a function. One example that comes to my mind is as follows: Suppose you are on a mountain and you want to get to the peak. We assume that I can climb to the peak however I want, in the sense that I am not restricted by terrain. All that matters is that following some path, I have to get to the peak. That path is a function. Now, for each path that I can take, I can apply some Zeno's paradox logic, so that my function can be associated with a series.

Victor Palea
  • 778