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Math Help - Trigonometric Limit

  1. #1
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    Trigonometric Limit

    Can anyone help me out with this question;

    Find the limit (as x approaches 0) of \frac{1 - \cos2x}{x^2}.
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  2. #2
    Super Member angel.white's Avatar
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    Quote Originally Posted by Flay View Post
    Can anyone help me out with this question;

    Find the limit (as x approaches 0) of \frac{1 - \cos2x}{x^2}.
    Using the identity cos(2x) = cos^2(x) - sin^2(x)

    \lim_{x\to 0} \frac{1-cos(2x)}{x^2}

    =\lim_{x\to 0} \frac{1-cos^2(x)+sin^2(x)}{x^2}

    =\lim_{x\to 0} \frac{sin^2(x)+sin^2(x)}{x^2}

    =2\lim_{x\to 0} \frac{sin^2(x)}{x^2}

    =2 \left( \lim_{x\to 0}\frac{sin(x)}x\right)^2

    Now \lim_{x\to 0}\frac{sin(x)}x Is well known to equal one. I was going to provide a proof from my book, but you really need to see the illustrations for it to make sense. Just know that it is proven with the squeeze theorem (and should be in your book as well, check where they talk about the squeeze theorem, and if you don't see it there, check where they start talking about derivatives of trigonometric functions. Then you can just say "according to theorem ### on page ###, \lim_{x\to 0}\frac{sin(x)}x=1"

    So, we apply the limit and get 1 for it and this becomes

    =2 \left( 1\right)^2

    =2
    Last edited by angel.white; April 8th 2008 at 10:27 AM.
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    Quote Originally Posted by angel.white View Post
    Now \lim_{x\to 0}\frac{sin(x)}x Is well known to equal zero.
    Of course, you meant to say 1.
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  4. #4
    Super Member angel.white's Avatar
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    Quote Originally Posted by Krizalid View Post
    Of course, you meant to say 1.
    Thank you, fixed it.
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