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Math Help - Trig limit help

  1. #1
    Member nautica17's Avatar
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    Trig limit help

    Where do I even start with this one?

    lim x -> 0 of ( (2 - cos(3x) - cos(4x) ) / x )

    Is there some identity that I can use that I'm not seeing.. ? I can't multiply by a conjugate can I?
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  2. #2
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    Quote Originally Posted by nautica17 View Post
    Where do I even start with this one?

    lim x -> 0 of ( (2 - cos(3x) - cos(4x) ) / x )

    Is there some identity that I can use that I'm not seeing.. ? I can't multiply by a conjugate can I?
    The slacker's way is to use l'Hopital's Rule.

    Otherwise you could substitute the Maclaurin series for cos(3x) and cos(4x), simplify the numerator and then cancel the common factor of x. Then take the limit.

    Alternatively, you could probably do a clever re-arrangement and get standard forms whose limits are well known. But why hoe the hard road?
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  3. #3
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    The simple little fact that \frac{1-\cos x}x\to0 as x\to0 will solve this:

    \frac{2-\cos 3x-\cos 4x}{x}=3\cdot \frac{1-\cos 3x}{3x}+4\cdot \frac{1-\cos 4x}{4x}. So the limit is zero.
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  4. #4
    Member nautica17's Avatar
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    Quote Originally Posted by Krizalid View Post
    The simple little fact that \frac{1-\cos x}x\to0 as x\to0 will solve this:

    \frac{2-\cos 3x-\cos 4x}{x}=3\cdot \frac{1-\cos 3x}{3x}+4\cdot \frac{1-\cos 4x}{4x}. So the limit is zero.
    That is an awesome trick! I give you kudos for that one. It's amazing how there so many ways to solve a math problem.
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  5. #5
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    Quote Originally Posted by nautica17 View Post
    Where do I even start with this one?

    lim x -> 0 of ( (2 - cos(3x) - cos(4x) ) / x )

    Is there some identity that I can use that I'm not seeing.. ? I can't multiply by a conjugate can I?
    If you were going to be a slacker (as I am ), you would notice that the limit tends to \frac{0}{0} by direct substitution.

    So you can use L'Hospital's Rule.

    \lim_{x \to 0}\frac{2 - \cos{3x} - \cos{4x}}{x} = \lim_{x \to 0}\frac{\frac{d}{dx}(2 - \cos{3x} - \cos{4x})}{\frac{d}{dx}(x)}

     = \lim_{x \to 0}\frac{3\sin{3x} + 4\sin{4x}}{1}

     = \frac{3\cdot 0 + 4\cdot 0}{1}

     = 0, as shown previously.
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