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Math Help - limit

  1. #1
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    limit

    What is the limit of x->1 sinpi*x/(1-x)^2?
    I want to solve it without using L'Hospital rule.
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  2. #2
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    Re: limit

    Hint: use the identity \sin(\pi-\theta)=\sin(\theta) and \lim_{x\to c}(f(x)\cdot g(x))=\lim_{x\to c}f(x)\cdot\lim_{x\to c}g(x) and a two-sided limit. What do you find?
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  3. #3
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    Re: limit

    sin(pi-x)*x*(pi-x)/(1-X)(1+x)(pi-x)=x*(pi-x)/(1-x)(1+x)?
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  4. #4
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    Re: limit

    No...I'm sorry but that makes no sense to me...please read my post above again and apply the identities I gave...
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  5. #5
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    Re: limit

    which one is f(x) or g(x)?
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  6. #6
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    Re: limit

    What is the answer then?
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: limit

    We are given to evaluate:

    \lim_{x\to1}\frac{\sin(\pi x)}{(1-x)^2}

    The first thing I am looking for is a way to apply the well-known result:

    (1) \lim_{u\to0}\frac{\sin(u)}{u}=1

    Seeing the denominator has 1-x as a factor, this takes me to \sin(\pi x)=\sin(\pi-\pi x)=sin(\pi(1-x)). So we now have:

    \lim_{x\to1}\frac{\sin(\pi(1-x))}{(1-x)^2}

    Now, let's use the substitution x=1-x and we have:

    \lim_{u\to0}\frac{\sin(\pi u)}{u^2}

    Now let's finish rewriting the expression so we can apply (1):

    \lim_{u\to0}\frac{\pi\sin(\pi u)}{\pi u^2}=\lim_{u\to0}\frac{\sin(\pi u)}{\pi u}\cdot\frac{\pi}{u}=\lim_{u\to0}\frac{\sin(\pi u)}{\pi u}\cdot\lim_{u\to0}\frac{\pi}{u}

    Applying (1), we are left with:

    \pi\lim_{u\to0}\frac{1}{u}

    Now, do we have:

    \lim_{u\to0^{-}}\frac{1}{u}=\lim_{u\to0^{+}}\frac{1}{u} ?
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  8. #8
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    Re: limit

    sin(pi*x) sin(pi*(x-1)+pi)
    --------- = ----------------
    (1-x)^2 (1-x)^2
    we can replace (1-x) with u.
    I am stuck in here.
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  9. #9
    MHF Contributor MarkFL's Avatar
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    Re: limit

    You have incorrectly applied the trigonometric identity.
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  10. #10
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    Re: limit

    Yes I found the answer.
    -sin(pi*(x-1))
    = --------------
    (1-x)^2

    sin(pi*(1-x))
    = ------------- ( since -sin(y) = sin(-y) )
    (1-x)^2
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  11. #11
    MHF Contributor MarkFL's Avatar
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    Re: limit

    Okay, but that's really unnecessary...did you actually read my post above where I showed you what to do except for the trivial last step? I hate to think my efforts are in vain.
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  12. #12
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    Re: limit

    I couldn't get it.I am sorry.
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  13. #13
    MHF Contributor MarkFL's Avatar
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    Re: limit

    What exactly about what I posted don't you get...I want to help, but my main objective is for you to truly understand...tell me where I lose you in my post...
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  14. #14
    MHF Contributor MarkFL's Avatar
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    Re: limit

    edit: web error...sorry.
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  15. #15
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    Re: limit

    answer is pi/2. How can I get this answer by 1/u?
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