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Math Help - Limits with trig values

  1. #1
    Newbie oObutterfly-chaserOo's Avatar
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    Question Limits with trig values

    I'm having trouble with my AP Calculus homework dealing with trig limits. I get these long values and have no idea how to reduce them.

    Example:

    Limit as x approaches zero of cot(3x)/csc(2x)

    I understand the first part:

    Lim x->0 (cos(3x)/sin(3x))/(1/sinx)

    which reduces to:

    Lim x -> 0 (cos(3x)sin(x))/sin(3x)

    but I'm at a loss of where to go from here. Everything I try either becomes so complicated I don't know what to do with it or ends up as 0/0. Thanks in advance to anyone who can shed some light on this dilemma.
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  2. #2
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    Quote Originally Posted by oObutterfly-chaserOo View Post
    I'm having trouble with my AP Calculus homework dealing with trig limits. I get these long values and have no idea how to reduce them.

    Example:

    Limit as x approaches zero of cot(3x)/csc(2x)

    I understand the first part:

    Lim x->0 (cos(3x)/sin(3x))/(1/sinx)

    which reduces to:

    Lim x -> 0 (cos(3x)sin(x))/sin(3x)

    but I'm at a loss of where to go from here. Everything I try either becomes so complicated I don't know what to do with it or ends up as 0/0. Thanks in advance to anyone who can shed some light on this dilemma.
    Hopefully you know that \lim_{x \to 0}\frac{\sin(ax)}{ax}=1 for all a \ne 0


    using this fact and some trig identities

    \frac{\cot(3x)}{\csc(2x)}=\frac{\cos(3x)}{\sin(3x)  }\cdot \sin(2x)

    Now multiply both the numerator and denominator by 6x to get

    \frac{6x}{6x}\frac{\cos(3x)}{\sin(3x)}\cdot \sin(2x)=\frac{2}{3}\cos(3x) \cdot \frac{3x}{\sin(3x)}\cdot \frac{\sin(2x)}{2x}

    using the above we get

    \lim_{x\to 0}\frac{2}{3}\cos(3x) \cdot \frac{3x}{\sin(3x)}\cdot \frac{\sin(2x)}{2x}=\frac{2}{3}\cdot 1 \cdot 1 \cdot 1=\frac{2}{3}
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  3. #3
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    Quote Originally Posted by oObutterfly-chaserOo View Post
    I'm having trouble with my AP Calculus homework dealing with trig limits. I get these long values and have no idea how to reduce them.

    Example:

    Limit as x approaches zero of cot(3x)/csc(2x)

    I understand the first part:

    Lim x->0 (cos(3x)/sin(3x))/(1/sinx)

    which reduces to:

    Lim x -> 0 (cos(3x)sin(x))/sin(3x)

    but I'm at a loss of where to go from here. Everything I try either becomes so complicated I don't know what to do with it or ends up as 0/0. Thanks in advance to anyone who can shed some light on this dilemma.
    L'Hospital's Rule works quite nicely here, since your limit is of the indeterminate form \frac{0}{0}.


    \lim_{x\to 0}\frac{\cos{(3x)}\sin{(2x)}}{\sin{(3x)}}

    = \lim_{x \to 0}\frac{2\cos{(2x)}\cos{(3x)} - 3\sin{(2x)}\sin{(3x)}}{3\cos{(3x)}}

     = \frac{2}{3}.
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