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

    where $\displaystyle [.]$ 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 $\displaystyle -1 < x < 0$ where

    $\displaystyle -1 < x^r < 0$ for $\displaystyle r$ odd $\displaystyle \Rightarrow [x^r ] = -1$
    $\displaystyle 0 < x^r < 1$ for $\displaystyle r$ even $\displaystyle \Rightarrow [x^r ] = 0

    $ so within this range

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

    Also in this range we have that

    $\displaystyle 1 + [x] + |x| + 2x = 1 -1 -x + 2x = x$

    and so since

    $\displaystyle \frac{0}{x} = 0$ when $\displaystyle x<0
    $

    $\displaystyle \lim_{x \to 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}= 0$
    Last edited by the_doc; Jun 12th 2009 at 09:16 AM. Reason: Changed > to < typo correction.
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