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Math Help - Continuity

  1. #1
    Junior Member
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    Continuity

    Hi everybody,

    I must show that the following function accepts a limit in 1 :

    f(x)=\frac{x^{p+1}-(p+1)x+p}{(x-1)^2} (p\in \mathbb{N*})

    I know that i should factorize by (x-1) but how?

    I need your help

    Thank's anyway.
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  2. #2
    MHF Contributor
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    France
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    Quote Originally Posted by lehder View Post
    Hi everybody,

    I must show that the following function accepts a limit in 1 :

    f(x)=\frac{x^{p+1}-(p+1)x+p}{(x-1)^2} (p\in \mathbb{N*})

    I know that i should factorize by (x-1) but how?

    I need your help

    Thank's anyway.
    Hi

    x^{p+1}-(p+1)x+p = x^{p+1}-1-(p+1)x+p+1

    x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k\right) -(p+1)(x-1)

    x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k -(p+1)\right)

    x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=1}^{p}x^k -p\right)

    Let g(x) = \sum_{k=1}^{p}x^k -p

    g(1) = \sum_{k=1}^{p}1 -p = 0

    Therefore g(x) can be factored by (x-1)
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  3. #3
    Junior Member
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    Quote Originally Posted by running-gag View Post
    hi

    x^{p+1}-(p+1)x+p = x^{p+1}-1-(p+1)x+p+1

    x^{p+1}-(p+1)x+p = (x-1)\:\left(\sum_{k=0}^{p}x^k\right) -(p+1)(x-1)

    ok thank you very much.
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  4. #4
    MHF Contributor
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    I have edited my post because you need to demonstrate that f can be factored by (x-1)
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  5. #5
    Junior Member
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    Quote Originally Posted by running-gag View Post
    i have edited my post because you need to demonstrate that f can be factored by (x-1) (wink)
    ok thank you again.
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