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Math Help - Limits (With a specified range?)

  1. #1
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    Limits (With a specified range?)

    Suppose 0≤ f(x) ≤1 for all x, find

    lim xf(x)
    x→0.


    This is exactly how the question is presented in my manual and I am not sure if it is trying to say that the function is x⁴ or something else. Also, I am not sure what relevance the first part of the question has to finding the limit. I believe it is representing the range, although I don't know what that means as far as limits go. Thank-you for your help.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Limits (With a specified range?)

    The function f(x) can't be x^4, because x^4 isn't surely bounded by [0,1] for all x, but if you rewrite:
    \lim_{x\to 0} x^4\cdot f(x)=\lim_{x\to 0}x^4\cdot \lim_{x\to 0} f(x)=... and you know 0\leq f(0)\leq 1 then the limit is ...
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    Re: Limits (With a specified range?)

    Thank-you for your quick reply.

    I am still a little unclear as to the part which states:

    0≤ f(x) ≤1 for all x.

    Does this represent the range of f(x)?

    I understand that if the limit of
    lim x⁴
    x→0

    is 0 that multiplying this by the limit of f(x) will yield a limit of 0. I am just wondering about the relevance of the range stated and whether that should affect my solution. Thank-you.
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  4. #4
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    Re: Limits (With a specified range?)

    Quote Originally Posted by Pewter12 View Post
    0≤ f(x) ≤1 for all x.
    Does this represent the range of f(x)?
    I understand that if the limit of
    lim x⁴x→0 is 0 that multiplying this by the limit of f(x) will yield a limit of 0.
    Actually this is the relevance of that condition.
    0\le f(x)\le 1 implies that 0\le x^4f(x)\le x^4.
    From that it follows at once that the limit is zero.

    We do not know that \lim _{x \to 0} f(x) = f(0)
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    Re: Limits (With a specified range?)

    I'm sorry, what I meant was that the product of

    lim x⁴ x lim f(x)
    x→0 x→0

    would be 0 because

    lim x⁴
    x→0 = 0

    Is there a particular rule telling us that
    0≤ f(x) ≤1

    implies also that

    0≤ x⁴f(x) ≤1?
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    Re: Limits (With a specified range?)

    Quote Originally Posted by Pewter12 View Post
    Is there a particular rule telling us that
    0≤ f(x) ≤1
    implies also that 0≤ x⁴f(x) ≤1?
    Well I did not say that it did: 0\le x^4f(x)\le x^4.
    However, if |x|<1 then x^4<1.
    So in the limit process that does in fact follow.

    But my point in post #4 is that you use the squeeze play.
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