limit / neighborhood, intorno (mathematics)

Marsario

Senior Member
Italia, italiano
Hi, I have been trying to translate the definition of a particular limit of a function. May anybody tell me whether what I have written is right?

In Italian it goes like this (sorry but I can neither use any images nor symbols here):
Limite +∞ di una funzione f(x) che tende a x_0:
Sia f una funzione definita in un intervallo [a ; b], escluso il punto x_0, si dice che la funzione f(x) tende a +∞ per x che tende a x_0 e si scrive " lim x-->x_0 f(x) = +∞ " quando per ogni numero reale positivo m si può
determinare un intorno completo I di x_0 tale che risulti f(x)>m∀x∈I ∩[a ; b]≠x0

My afford:
Limit +∞ of a function f(x) which approaches to x_0
Suppose f a function defined in an interval [a;b] except the point x_0, we say that the limit of f(x) is +∞ as x approches to x_0 and write:
lim x-->x_0 f(x) = +∞
if and only if for every real m > 0 there exists a neighbourhood I of x_0 such that it implies f(x)>m∀x∈I ∩[a ; b]≠x0

Thank you very much in advance for your help!
 
  • brian

    Senior Member
    AmE (New Orleans)
    Your try is pretty good! But I will put it into more mathese for you.

    Marsario said:
    Limite +∞ di una funzione f(x) che tende a x_0:
    Sia f una funzione definita in un intervallo [a ; b], escluso il punto x_0, si dice che la funzione f(x) tende a +∞ per x che tende a x_0 e si scrive " lim x-->x_0 f(x) = +∞ " quando per ogni numero reale positivo m si può
    determinare un intorno completo I di x_0 tale che risulti f(x)>m∀x∈I ∩[a ; b]≠x0

    The limit +∞ of a function f(x) which approaches x_0:
    Let f be a function defined over an interval [a,b], excluding the point x_0. We say that the function f(x) tends to +∞ as x tends to x_0 and we write "lim x-->x_0 f(x) = +∞" if and only if* for every real number m>0, there exists a neighborhood** I of x_0 such that f(x) > m ∀x∈I ∩ [a,b] ≠ x_0.


    *if and only if actually means se e solo se; quando ("when") is mathematically less strong. But in this case, if and only if is mathematically correct, so I suppose using it as a translation is fine.

    **I don't know what an intorno completo is. How is it different from a normal intorno/neighborhood?
     

    Marsario

    Senior Member
    Italia, italiano
    Thank you very much Brian! Yours was a great help!
    Actually I am not sure myself about the difference between Intorno completo and Intorno.
    Basically besides Intorno completo there are at least Intorno circolare, Intorno destro and Intorno sinistro.
    Although I know the definitions of these last neighbourhoods I have never heard the definition of a common Intorno.
    Anyway I suppose it is enough using the world Neighbourhood in my definition, surely it suits what I need it for.
    Thanks again!
     

    elemika

    Senior Member
    Russian
    La parte di un intorno completo di x0 che sta a destra di x0 si dice intorno destro I(x0+) di x0 .La parte di un intorno completo di x0 che sta a sinistra di x0 si dice intorno sinistro I(x0-) di x0.
    So, we have left neighborhood, right neighborhood and full neighborhood
     
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