French limits are different !

Question

Well, well, well, back in the 1980's france used to show 12 grade student the =definition of limits through adherent points and made them prove it was equivalent to Def.1 below. (yes you read that right, 12 grade) and prove them lots of theorem about them. They had to prove the composition theorem!. But, in 1988, when they changed the mathematical books, they did a small change to the definition of limit, that today is still taught :

Def 1.

for non-french people or french people from before the change$$\lim\limits_{x\to x_0} f(x) = l \Leftrightarrow \left(0<|x-x_0|<\alpha \implies|f(x)-l|<\varepsilon\right)$$

Def 2.

Baguette definition (falsely alledged to Bourbaki) $$\lim\limits_{x\to x_0} f(x) = l \Leftrightarrow \left(|x-x_0|<\alpha \implies|f(x)-l|<\varepsilon\right)$$ Or in good ol' French

On dit que la fonction $f$ définie en $x_0$ admet pour limite $l$ en $x_0$ si et seulement si, lorsque la distance entre $x$ et $x_0$ est inférieure à $\alpha$, nombre positif aussi petit que l'on veut, la distance entre $f(x)$ et $l$ est inférieure à $\varepsilon$ ce dernier ayant la même définition que $\alpha$.

with $\alpha$ and $\epsilon$ being positive, $f$ is of course defined at point $x$ and $x_0$

The proof of the limit composition theorem becomes easier with Def 2. This is why they changed it.

What is simpler/easier with Def 2. ?

This I cannot figure out. Thanks in advance for helping me! Tom

Answer 1

The absolute value can now become $0$. I believe that might imply that $f$ has to be defined in $x_0$, but that would be a matter of context...

If $f$ has to be defined in $x_0$ there will be big trouble with extending a continuous function's domain by using the limit (see sinc-function).

But it is soothing that at least the french still learn some math at school. I should migrate...

EDIT and remark: (for future visitors)

An assumption of $x_0$ being in the domain of $f$ may or may not be made at school, this I cannot assess.

From Siminore's answer and from the comments to the question, linking the french wikipedia, it has however emerged that the specific definition of a limit where $x_0$ is indeed part of the domain of $f$ can be helpful with respect to proving statements about the limit of the composition of several functions.

Answer 2

I have met a colleague in Brescia who teaches limits the french way. In his notes he says that the only reason to prefer the french definition is that the limit of compositions becomes easier. I do not believe that there are other reasons, actually.

Answer 3

What is simpler/easier with Def 2. ?

Maybe the relevant issue is pedagogic.

I suspect that, for 12 grade students (which is the context of your question), "functions defined on some open interval that contains $x_0$" are easier to handle (conceptually) than "functions defined on some open interval that contains $x_0$, except possibly at $x_0$ itself".

The biggest "problem" of Definition 2 is that, according to it, a function cannot have a limit at a discontinuity point of its domain. But this is not a real problem if our analisys considers only continuous functions (which would be reasonable for 12 grade students).