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Trigonometry limit

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data
    [tex]\lim_{x \to \frac{\pi}{3}} \frac{1-2 cos x}{\pi - 3x}[/tex]



    2. Relevant equations
    trigonometry identity
    properties of limit for trigonometry

    3. The attempt at a solution
    I have done several attempts but got me nowhere. I just need an idea to start.

    Thanks
     
  2. jcsd
  3. May 5, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    Do you know the rule of l'Hospital?
     
  4. May 5, 2013 #3
    Yes and I am not allowed to use it.
     
  5. May 5, 2013 #4

    mfb

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    2016 Award

    Staff: Mentor

    Okay. Can you use a Taylor series?
    Without any derivatives or approximations to the functions, it looks tricky.
     
  6. May 5, 2013 #5
    I haven't learnt it yet. I think I am only allowed to use trigonometry identities and limit properties
     
  7. May 5, 2013 #6
    Write [itex]\lim_{x \to \frac{\pi}{3}} \frac{1-2 cos x}{\pi - 3x}[/itex] as [itex]\frac{2}{3}\lim_{x \to \frac{\pi}{3}} \frac{1/2- cos x}{\pi/3 - x}[/itex]. Notice that if you replace 1/2 with cos(π/3) you get something that looks like the definition of a derivative. It should be 2/3*cos'(π/3)

    Edit: Do you know the derivative of cosine? If not, it is easy to calculate if you know the (1-cosx)/x and sinx/x limits.
     
  8. May 5, 2013 #7
    I get it. Thanks a lot for your help :smile:
     
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