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Math Help - Greatest integer and modulus function limit?

  1. #1
    Super Member fardeen_gen's Avatar
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    Greatest integer and modulus function limit?

    Evaluate:
    \lim_{x\rightarrow 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}

    where [.] denotest the greatest integer function.

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  2. #2
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    I don't believe this function's limit can be defined from the left as the greatest integer function jumps from when x is just less than 0 to when it is 0.
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  3. #3
    Moo
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    Quote Originally Posted by the_doc View Post
    I don't believe this function's limit can be defined from the left as the greatest integer function jumps from when x is just less than 0 to when it is 0.
    Yet it is still defined as x doesn't technically attain 0... ?
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  4. #4
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    Had a look at the question again and you're correct Moo - I misread. It does exist as we only need to get arbitrarily close to x=0 to define the limit.

    In which case we can consider the range -1 < x < 0 where

    -1 < x^r < 0 for r odd \Rightarrow [x^r ] = -1
    0 < x^r < 1 for r even \Rightarrow [x^r ] = 0<br /> <br />
so within this range

    \sum_{r=1}^{2n+1} [x^r ] = -(n+1) .

    Also in this range we have that

    1 + [x] + |x| + 2x = 1 -1 -x + 2x = x

    and so since

    \frac{0}{x} = 0 when x<0<br />

    \lim_{x \to 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}= 0
    Last edited by the_doc; June 12th 2009 at 10:16 AM. Reason: Changed > to < typo correction.
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