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Math Help - Proving the limit...

  1. #1
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    Proving the limit...

    I get how to get the answer, but I need to use formal definition of limits... something I am very unfamiliar with.

    lim x->a g(x) = infinity and g(x) < or equal f(x) for x -> a


    I know the answer is lim as x->a for f(x) = infinity... but what definitions do I need to do this? I am not sure how to show work for this question.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by nein12 View Post
    I get how to get the answer, but I need to use formal definition of limits... something I am very unfamiliar with.

    lim x->a g(x) = infinity and g(x) < or equal f(x) for x -> a


    I know the answer is lim as x->a for f(x) = infinity... but what definitions do I need to do this? I am not sure how to show work for this question.
    Well, share the question with us, and maybe we can help.
    Last edited by VonNemo19; June 17th 2009 at 05:20 PM. Reason: Edited my edit reason
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  3. #3
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    Quote Originally Posted by VonNemo19 View Post
    Well, share the question with us, and maybe we can help.
    Oops... my bad.

    If lim x->a g(x) = infinity and g(x) < or equal to f(x) for x -> a, then limx-> a f(x) = infinity

    Prove, using the formal definition of limits, each of the statement is true.

    So... that's the question. Honestly, I never had to answer this kind of questions. It's usually applications format or equation format.

    Another one is limx->infinity f(x) = -infinuty and c > 0, then limx ->infinity c f(x) = - infinity

    Not sure which rules to use, but I can reason... that f(x) is a negative infinity... times bu any constant positive number is still negative infinity. I am not sure if there are any "specific rules".
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  4. #4
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by nein12 View Post
    Oops... my bad.

    If lim x->a g(x) = infinity and g(x) < or equal to f(x) for x -> a, then limx-> a f(x) = infinity

    Prove, using the formal definition of limits, each of the statement is true.

    So... that's the question. Honestly, I never had to answer this kind of questions. It's usually applications format or equation format.

    Another one is limx->infinity f(x) = -infinuty and c > 0, then limx ->infinity c f(x) = - infinity

    Not sure which rules to use, but I can reason... that f(x) is a negative infinity... times bu any constant positive number is still negative infinity. I am not sure if there are any "specific rules".
    So we wish to show that if \lim_{x\to{a}}g(x)=\infty and g(x)\leq{f(x)} as ? x\to{a} ?, then \lim_{x\to{a}}f(x)=\infty

    Well, for infinite limits the definition states:
    Let g(x) be a function that is defined on an interval containing a, except possibly at a. Then we say that

    \lim_{x\to{a}}g(x)=\infty

    if for every number M>0, there is some number \delta>0 such that
    g(x)>M whenever 0<\mid{x-a}\mid<\delta

    So then the proof is straight forward

    if g(x)>M for every x arbitrarily close to a, then f(x)>g(x) for every value of x arbitrarily close to a \Rightarrow{f(x)>M}

    Therefore
    f(x)>M whenever 0<\mid{x-a}\mid<\delta
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  5. #5
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    Quote Originally Posted by VonNemo19 View Post
    So we wish to show that if \lim_{x\to{a}}g(x)=\infty and g(x)\leq{f(x)} as ? x\to{a} ?, then \lim_{x\to{a}}f(x)=\infty

    Well, for infinite limits the definition states:
    Let g(x) be a function that is defined on an interval containing a, except possibly at a. Then we say that

    \lim_{x\to{a}}g(x)=\infty

    if for every number M>0, there is some number \delta>0 such that
    g(x)>M whenever 0<\mid{x-a}\mid<\delta

    So then the proof is straight forward

    if g(x)>M for every x arbitrarily close to a, then f(x)>g(x) for every value of x arbitrarily close to a \Rightarrow{f(x)>M}

    Therefore
    f(x)>M whenever 0<\mid{x-a}\mid<\delta
    Umm... I am very good with math. What is M and delta mean?
    And I've never seen 0<\mid{x-a}\mid<\delta expression before, though I kinda understand it.
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  6. #6
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by nein12 View Post
    Umm... I am very good with math. What is M and delta mean?
    And I've never seen 0<\mid{x-a}\mid<\delta expression before, though I kinda understand it.

    The problem you posted requires knowlege of the epsilon-delta (or formal) defintion of limit.

    M is some arbitrarily large number, loosely speaking, delta is the distance x is away from a, and the inequality is a way of narrowowing down the distance between a and x to any desired length.
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