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Math Help - Trigonometric Limits

  1. #1
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    Trigonometric Limits

    Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.


    1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

    2. lim (sin(3t))/(2t) as t approaches 0

    3. lim (sin(2x))/(sin(3x)) as x approaches 0


    Any help would be appreciated!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Enderless View Post
    Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.


    1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

    2. lim (sin(3t))/(2t) as t approaches 0

    3. lim (sin(2x))/(sin(3x)) as x approaches 0


    Any help would be appreciated!
    what have you tried so far?

    for the first, change everything to be in terms of sine and cosine and see what cancels out

    for the second, use the special result. \lim_{x \to 0} \frac {\sin x}{x} = 1

    for the third, expand the sines with the double angle formula and see what cancels out.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Enderless View Post
    Hi all, I am having trouble simplifying or factoring the following trigonometric limits. I can't find the identities for the last two.


    1. lim (1-tan(x))/(sin(x)-cos(x)) as x approaches pie/4

    2. lim (sin(3t))/(2t) as t approaches 0

    3. lim (sin(2x))/(sin(3x)) as x approaches 0


    Any help would be appreciated!
    2. and 3. are trivial just expand the sin's as power series (you will just need the first non zero term).

    RonL
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  4. #4
    MHF Contributor red_dog's Avatar
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    For 2 and 3 use this:
    \displaystyle\lim_{x\to 0}\frac{\sin\alpha x}{\beta x}=\frac{\alpha}{\beta} and \lim_{x\to 0}\frac{\sin\alpha x}{\sin\beta x}=\frac{\alpha}{\beta}.

    Proof:
    \lim_{x\to 0}\frac{\sin\alpha x}{\beta x}=\lim_{x\to 0}\frac{\sin\alpha x}{\alpha x}\cdot\frac{\alpha}{\beta}=\frac{\alpha}{\beta}.

    \lim_{x\to 0}\frac{\sin\alpha x}{\sin\beta x}=\lim_{x\to 0}\frac{\sin\alpha x}{\alpha x}\cdot\frac{\beta x}{\sin\beta x}\cdot\frac{\alpha}{\beta}=\frac{\alpha}{\beta}.
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  5. #5
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    Going over my work, I've found the solutions to the problems already before Jhevon posted a reply. It turns out that it was easier than I thought by using the triple angle identities or something like that.

    Thanks for the replies everyone!
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Enderless View Post
    Going over my work, I've found the solutions to the problems already before Jhevon posted a reply. It turns out that it was easier than I thought by using the triple angle identities or something like that.

    Thanks for the replies everyone!
    Good job! and good luck with your class
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