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Math Help - Continuity with limits

  1. #1
    Ife
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    Continuity with limits

    I am having some trouble with the concept of continuity and limits when it applies to questions. Here's a simple question I am not sure how to approach:

     \mbox{If}   x^4 \leq \mbox{f(x)} \leq x^2 for -1 \leq x \leq 1 and x^2 \leq \mbox{f(x)} \leq x^4 for x<-1 \mbox{or} x>1, at what points \mbox{c} do you automatically know \lim_{x\to\mbox{c}}\mbox{f(x)}?

    What are the values of the limits at these points?

    What is the correct approach to this question?
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  2. #2
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    Use sandwich theorem.

    In x = 0, -1, 1 . You can tell what the limits are.

    In x = 0, the limit is zero.
    In x = 1, -1 the limit is one.

    It is a consequence of the sandwich theorem:

    \lim_{x \to 0} x^{4} = \lim_{x \to 0} f(x) = \lim_{x\to 0} x^{2}

    the same can be said in x = -1,1 though in these cases you have to consider lateral limits.
    Last edited by Diego; April 16th 2010 at 01:46 PM.
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  3. #3
    Ife
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    Quote Originally Posted by Diego View Post
    In x = 0, -1, 1 . You can tell what the limits are.

    In x = 0, the limit is zero.
    In x = 1, -1 the limit is one.

    It is a consequence of the sandwich theorem:

    \lim_{x \to 0} x^{4} = \lim_{x \to 0} f(x) = \lim_{x\to 0} x^{2}

    the same can be said in x = -1,1 though in these cases you have to consider lateral limits.
    Thanks.. though i am not sure I know about Lateral Limits...
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  4. #4
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    I believe he means "one-sided" (from the left and from the right) limits.
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  5. #5
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    Lateral limits

    Oh, sorry I meant one-sided limits (in spanish they are called lateral). For instance in x=1, if you approach the limit with numbers bigger than one (where x^{2} < f(x) < x^{4}):

    \lim_{x \to 1^{+}} x^{2} = \lim_{x \to 1^{+}} f(x) = \lim_{x\to 1^{+}} x^{4}=1

    and if smaller than one (where x^{4} < f(x) < x^{2}, if x is not too far from 1):

    \lim_{x \to 1^{-}} x^{4} = \lim_{x \to 1^{-}} f(x) = \lim_{x\to 1^{-}} x^{2} =1

    thus:


     \lim_{x \to 1} f(x) = 1
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