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Math Help - deleted neighborhood of limit L

  1. #1
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    deleted neighborhood of limit L

    The question is this true or false. Justify.

    Let f:[a,b]-> \Re and let c \in (a,b). If lim_{x->c} f(x)=L and L>0, then there is a deleted neighborhood N*^(c) such that f(c)>0 for all x \in N*(c).

    If this is true, is the reason because of the following theorem.

    Let f: D-> \Re and let c be an accumulation point of D. Then lim_{x->c} iff for each neighborhood V of L there exists a deleted neighborhood U of c such that f(U \capD) \subseteq V.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: deleted neighborhood of limit L

    Right. We only have to choose V=(0,2L) .
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    Re: deleted neighborhood of limit L

    If I understand your terminology correctly, I think that using anything more than the definition of the limit is overkill. If f(x)->L as x->c, and L>0, then choose epislon = L/2. This is a standard type of scenario, both the problem and its solution, and so this standard approach is the best way to do it.
    By the definition of the limit, when epsilon = L/2 (note it's positive), there's some positive delta that defines a corresponding deleted open neighborhood U of c
    ( U = [a,b] intersect (c-delta, c+delta) intersect {c}-complement )
    such that for all x in U, abs(f(x)-L) < L/2. Thus for all x in U, -L/2 < f(x)-L, and thus f(x) > L/2. Since L/2 > 0, have shown that there exists a deleted open neighborhood U of c such that for all x in U, f(x) is positive.
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    Re: deleted neighborhood of limit L

    Quote Originally Posted by johnsomeone View Post
    If I understand your terminology correctly, I think that using anything more than the definition of the limit is overkill. If f(x)->L as x->c, and L>0, then choose epislon = L/2. This is a standard type of scenario, both the problem and its solution, and so this standard approach is the best way to do it.
    By the definition of the limit, when epsilon = L/2 (note it's positive), there's some positive delta that defines a corresponding deleted open neighborhood U of c
    ( U = [a,b] intersect (c-delta, c+delta) intersect {c}-complement )
    such that for all x in U, abs(f(x)-L) < L/2. Thus for all x in U, -L/2 < f(x)-L, and thus f(x) > L/2. Since L/2 > 0, have shown that there exists a deleted open neighborhood U of c such that for all x in U, f(x) is positive.
    Note that 0 < \left| {x - c} \right| < \delta is a deleted neighborhood (x-\delta,x)\cup(x.x+\delta) of x.
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  5. #5
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    Re: deleted neighborhood of limit L

    Yes. I wrote the deleted nbhd part as "(c-delta, c+delta) intersect {c}-complement" to avoid a more complex statement containing both unions and intersections when full description of the set included the domain [a,b]. That's why I wrote it as "U = [a,b] intersect (c-delta, c+delta) intersect {c}-complement". I thought it might be better than "[a,b] intersect ( (c-delta,c) union (c, c+delta) )", but it's probably not. It's probably more confusing.
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