Using the geometric (infinitesimal) approach suggested by Efremovich during the early 1930s, we can write

A function f:X -> R has a limit L at p, provided f(x) is close to L, eventually.

The word "close" conveys the geometric (spatial) interpretation of f(x) being close to L. It was Smirnov, Efremovich's student, who walked away from the geometic interpretation of closeness in his introduction (in the early 1950s) to Efremovich's view of proximity. It was Leader, Naimpally and others who began using the term "near" instead of "close" to convey the proximity of a point to a set (topology) or the proximity of one set to another set (proximity space theory).

The best techniques will be the numerical approximation which I used in lectures. That´was the way I learned it and later I understood the limit intuitively.

Dear Abedallah,

sometimes my students found it helpful to look in youtube for some educational videos. Nowadays practically all subjects are somehow explained in youtube videos. For example when you search for "limit of a function at a point" in youtube you will easily find some helpful explanation for students - and students like youtube and the internet very much!!!

Regards Reiner Creutzburg

Dear Abedallah,

In order to explain the concept of the limit to students one should have in his mind what is their specialization and how much time they are eager to dedicate to mathematics. In any case it should be mentioned that the concept of the limit is very basic and one needs to return to it many times during his study.

I remember perfectly well my own doubts when I was studying this concept and usually try to explain the fool questions that bother me at that time. It is also very important to push your students to tell you the fool questions which they have when you are explaining the concept.

If we have to use the strict {epsilon,delta} definition, then we should use a good diagramm too. Otherwise, we have to emphasize the fact that limit means "as near as we desire", by a suitable and easy way. It depends on the Department.

Dear @Hanno, Numerical approximation is a very good technique to explain the concept of the limit; sometimes the background of the students is not good enough to concentrate on numerical approximation first and thereafter the limit.

Dear @Reiner, The youtube is a very good resource to explain the concept of the limit.

Dear @Kazaros, I also push my students to tell me the fool questions which they have when I am explaining the concept of the limit; it leads to interactive learning for the students and the fool ideas about the concept of the limit can be explained.

Dear @Demeris, I agree with you that the main idea is based on emphasizing the fact that limit means as near as we desire.

Dear @Abedallah, Maybe some kind of analogy is helpful. I remember in learning FEM I worked out how to find the circumference of a circle without using the formula P=2piR. we represented the circle by a number of cords. Their combined length (adding the straight lines) gave P. Now If we divide the circumference of the circle into n cords, the total length of the cord and the approximate value of P is obtained. To get the exact value, we will apply the L’Hopital Rule, As the number of cords approaches infinity (limit as n--> infinity), the value of the P approaches the exact value of 2piR.

Dear @Mahmoud, It is a good idea to give the students an example to calculate the limit using calculations and conclude the limit.

Dear @Esteven, It is a very good hint to use a real time software to explain the limit.

As mathematician is very difficult for me to define something without the necessary precision and the use of symbols. However, in a simple way, the concept of limit can be explained in the following manner:

Let consider the function f (x) defined on the whole interval [a, b]. Consider the point c within the interval [a, b] and be f(c) the value of the function in point c. The limit of the function f (x), when the point x move toward the point c, is the value f(c).

You can use the graphic of the function f (x) to explain in a practical manner the concept of limit. You can do it by plotting the interval [a, b] and the point c in the axis X and then explain that when the point x move towards the point c the points in the graphic of f(x) move toward the point f(c). Of course you have to explain later on, when the concept of limit is more clear to the students, all possible alternatives that are associated with this concept.

A research (see attached link) showed using computers has a positive effect in visualization for teaching limit concept. The purpose of their study was to discover if the students can comprehend the limit concept and overcome its difficulties by the help of computer.

http://www.cimt.plymouth.ac.uk/journal/buyukkoroglu.pdf

Dear @Jorge, Thank you for the contribution; It is interesting:

Let the function f (x) be defined on the whole interval [a, b] and the point c within the interval (a, b); then the limit of the function f (x) equals L, when the point x move toward the point c, if accordingly the value of f(x) approaches L.

Dear @Mahmoud, Thank you for the article; it contains interesting results about using MATLAB in explaining the concept of the limit. The study divides a group of Math students into 2 groups A and B; group A taught using the traditional method, and the second group B taught using the computer. Then both groups took an exam and the performance of group A was 35% and group B was 46%.

Dear Abedallah. You can use L instead of f(c) for the limt of the function f(x). This is also correct and I used it many time in the past.

Dear @Jorge, Yes, the symbols L and f(c) can be used; however, since f(c) refers to the value of f at c and for the limit f(c) need not to be defined; this means that we implicitly assume that the function f is continuous.

Dear Abedallah. During the definition of the limit the student has no idea of the concept of continuity of a function. For this reason, students will not detect this. You can explain all types of limits and later on the concept of continuity clarifying the differences between these two concepts, but after the student has a clear picture of what the limit is.

The use of a point L without any clear relationship with the function f(x) could be an additional difficulty that the student should overcome in understanding the concept of limit.

I use L or f(c) depending on the level of the students that I have in the class.

Dear @Jorge, Thanks for the feedback. This is a good point. Yes, we implicitly assume that the function f is continuous, so we can use both approaches, in general. Like you, I use levels of difficulty to explain the concept of limit because the students do not need additional level of difficulty.

It is possible to define the limit of a function at a point using the near set approach. That is, a function f(x) has a limit L at p, provided f(x) is near L, eventually.

My point is that we must explain the limit of a function as simple as possible avoiding, at the begining, to add to much complication to the explanation. When the concept is well understood by the students using examples and the graphic of the function, then we can introduce more and more elements in the definition in other to allow the students to fully understand the concept using a mathematic expression. If students have problem in understand the concept of limit, then they will have more a more problems in understand the concept of continuity and derivatives.

Dear @Jorge, As you said, in most limit problems the first step should be to graph or otherwise visualize the function on an appropriate interval. However, a graph is not always sufficient to determine a limit. If the function is defined at the limit point and there are no "problems" at the limit point, we can identify limits this way (graphically).. When we have division by zero or other peculiarities in the function it can be misleading.

Using the geometric (infinitesimal) approach suggested by Efremovich during the early 1930s, we can write

A function f:X -> R has a limit L at p, provided f(x) is close to L, eventually.

The word "close" conveys the geometric (spatial) interpretation of f(x) being close to L. It was Smirnov, Efremovich's student, who walked away from the geometic interpretation of closeness in his introduction (in the early 1950s) to Efremovich's view of proximity. It was Leader, Naimpally and others who began using the term "near" instead of "close" to convey the proximity of a point to a set (topology) or the proximity of one set to another set (proximity space theory).

Dear Mahmoud. You can complicate the explanation of the concept of limit as much as you wish, but this approach will not make easier for the student the understanding of the concept. Of course that you can use a very complicate graphic of a function and use it to explain the concept of limit in all of its form, but this will not help. My approach is different. Use a graphic of a simple and easy function and explain the concept of limit and when the student understand the concept, then begin to present all posible forms of the limit, this measn limit to the right, to the left, of a function not defined in point c and so on and so on.

Dear Jorge. I agree with you. we should start with simple functions and work out few simple examples and graph them. Then we move to those functions with "problems".

Thank you dear @James for bringing this definition under light: A function f(x) has a limit L at p, provided f(x) is near L, eventually.

Dear @Mahmoud and @Jorge, Yes it depends on the level of the students and their study directions. We wish that the students are capable of understanding the formal definition with some explanations using words and/graphs but sometimes not possible with vane.

Dear @Abedallah and colleagues, what do you think about this approach in limit of a function explanation!? Spend 5 minutes and comment!

http://www.youtube.com/watch?v=HYSI-AHUqRM

How to explain the formal definition of limit in simple words? The formal definition of limits is: The limit of the function f(x) at the point a is L if and only if for any epsilon > 0 there exists delta > 0 such that if 0 < | x - a | < delta then |f(x) - L| < epsilon. In my practice I just change the definition a little bit:

The limit of the function f(x) at the point a is L if and only if for all epsilon > 0 (Notice that it doesn't matter how small epsilon is.) there exists delta > 0 such that if 0 < | x - a | < delta then |f(x) - L| < epsilon. In other words: we can make the value of the function as close to L as we want if we make the argument of f(x) as close to a as required.

In my experience, most of the students have difficult to understand the formal definition of limit, if you do not explain it in the most simple and practical manner, unles you are teching students in the Mathematic's Faculty. The formal definition of limit can be introduced once the students have a clear idea what the concept of limit is. I advice to use the formal definition of limit to solve practical problems in order to allow the students to understand better the formal definition of this important concept.

Dear Ljubomir, I saw the small video and it seem to me the use of that this video is not going to help to understand the concept of limit, but I could be wrong.

Although the word "simple" is a bit ambiguous, the formal ε-δ definition of limit of a function f at x = c is the best possible way of describing the phenomena in the language of mathematics and the intuitive description of it is to replace the ε-δ closeness by words - arbitrary close or as close as we please my serve the job.

Besides graphs are good tools to display what we describe in symbols and convey the concepts we want to teach for our students : we say L is the limit of f(x) at x = c if for every non empty near set of L, there is a non empty near set about x = c that need not necessarily contain x = c such that all points from the near set of x = c are mapped to the near set of L we have picked - in a sense near points of x = c are mapped to near points of L except possibly x = c.

It is also good and helpful to show them: what then makes a number not to be a limit of a function at x = c ( or why limit fails) - if when for a selected non empty near set of L, and for every non empty near set of x = c, there are points in the near set of x = c which are mapped not to be near L or mapped to stay away from being near L - this will indirectly throw a light on their grasp and understanding on the concept of what a limit of a function at a point indeed mean.

But above all: Euclid is believed to have said - " no royal road to geometry" for King Ptolemy 's request of easy way of learning mathematics.

Thank you dear @Ljubomir for the interesting video; it is interesting how he explained the limit by taking the of y=x-4 as x->5.

Dear @Oscar, This is the way I explain the limit at my university, and it is okay. My experience in other universities, north america: it is not easy for the students to understand this definition.

Yes dear @Dejenie: "no royal road to geometry".

The over simplifications on the definition may also lead to a wrong understanding of what limit of a function at a point is by beginner students. Some how the quantifiers we use to define the concept should be included in the simplified descriptions. For instance some of the exercises in some calculus text books are to figure out what is wrong with such simplified descriptions of a definition of a limit. Students will take those descriptions literally correct, but deep in their descriptions they lack important components, particularly of quantifies.

Dear Abedallah,

I would like to add the following to my previous post. For students which are not mathematicians I prefer to explain the concept of limit for sequences and afterwards define the limit of a function in some point in terms of sequences. I guess there is no need to formulate it here.

Regards,

@Kazaros

That's Heine's definition of a limit. I assumed that the question referred to Cauchy's definition, but you are right, that's another approach. The main source of confusion, I think, is the quantifier "for all epsilon" in Cauchy's definition.

Dear Oscar,

The point is that in this way one can avoid to use terminology epsilon-delta just using properties, lemmas and theorems about limits of sequences. But epsilon is there in both cases.

Using Non Stndard Analysis, the definition is very simple. let f a standard function and x0 in R a standard element. Let a an infinitesimal element of R*. The limit of f in x0 is the standard part of f(x0+a). But you have to introduce the valued non archimedian Robinson extension R* of the real field R. But it is not more complicated that the definition of the set R.

Dear @Dejenie, an interesting point: the simplification should reflect the values of the formal definition of the limit and should not lead to a false understanding by the students.

Dear @Kazaros, this is a good strategy and simplifies the explanation, assuming that the students did study sequences or sequences are covered by the course. Thereafter as dear @Oscar pointed out: the Delta concept should be included accordingly.

Dear @M. Goze, thank you for introducing a new terms in the discussion.

http://www.raco.cat/index.php/ensenanza/article/viewFile/21683/21517

Dear @Jose, thank you for the interesting article; do you have the article in English.

The Abstract of the article: Our research falls within the general field of the analysis of the teacher’s activity and focuses on the specific case of

teaching the concept of «limit to the function» in the Spanish secondary school system. Using the anthropological focus

of didactics (Chevallard, 1998) as a general theoretical frame, we propose an investigative methodology for the

analysis of mathematical organizations recreated by the teacher in the classroom in collaboration with his/her pupils

and the respective didactic organizations that allow their reconstruction.

Dear José. Many thanks for the link with the article on how to teach the concept of limit of a function in the secondary school. The article is very interesting from the methodological point of view, but provide no direct answer to the main subject of the debate.

No tengo en ingles el documento,...el tradurctor de red...

Se que no resuelve la cuestión de fondo, pero si apropiarse de un concepto es evolucionar hacia su visión científica, creo que el camino conlleva evolución, aunque sea de visión errónea en visión erronea... es decir, los que importa es que evolucione su sistema de ideas, su cosmovisión del concepto, y no tanto su llegada.

Por eso es más interesante, para el desarrollo profesional y para la evolución de las ideas de los estudiantes, la METODOLOGÏA

No tengo en ingles el documento,...el tradurctor de red...

Se que no resuelve la cuestión de fondo, pero si apropiarse de un concepto es evolucionar hacia su visión científica, creo que el camino conlleva evolución, aunque sea de visión errónea en visión erronea... es decir, los que importa es que evolucione su sistema de ideas, su cosmovisión del concepto, y no tanto su llegada.

Por eso es más interesante, para el desarrollo profesional y para la evolución de las ideas de los estudiantes, la METODOLOGÏA

Dear @Jose, thank you for the feedback; I can not Spanish but I used a translator.

Dear @George, thank you for your contribution; the formal definition of the limit is given but it is somehow difficult for the students to understand, so we are seeking for simple method and words to describe it.

Dear @George, I think this is the golden approach that most of us follow to explain the limit; however, there is always the difficulty of the \epsilon-\delta explanation.