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Math Help - THEORY: Why the definition of limit requires an open interval?

  1. #1
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    THEORY: Why the definition of limit requires an open interval?

    Hello,

    although I understand what limit is about I don't understand why the definition requires an interval like this

    \left ( a,p \right )\cup \left ( p,b \right )

    and not one like this:

    \left [ a,p \right )\cup \left ( p,b \right ]

    when it wants to show that there exists the limit on the point p. How the second one would harm the limit definition?

    Thank you in advance!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: THEORY: Why the definition of limit requires an open interval?

    Quote Originally Posted by Melsi View Post
    \left ( a,p \right )\cup \left ( p,b \right ) and not one like this: \left [ a,p \right )\cup \left ( p,b \right ]
    It is irrelevant. In both cases, the function is defined for points sufficiently close to p, so it has sense to study the behavior of f(x) as x\to p . Limit is a local concept.
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    Re: THEORY: Why the definition of limit requires an open interval?

    Quote Originally Posted by Melsi View Post
    Hello,

    although I understand what limit is about I don't understand why the definition requires an interval like this

    \left ( a,p \right )\cup \left ( p,b \right )

    and not one like this:

    \left [ a,p \right )\cup \left ( p,b \right ]

    when it wants to show that there exists the limit on the point p. How the second one would harm the limit definition?

    Thank you in advance!
    Dear Melsi,

    As FernandoRevilla had said it is irrelevant and would not harm the limit definition.
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  4. #4
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    Solved

    I understand, you mean it doesn't use the closed one because it wants to show that it is not interested on a, b themself but on those point going very close to p.

    It seems logical, I just was afraid that might be something else that I missed out.

    Thank you very much for making it clear.
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