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Math Help - Limit

  1. #1
    Junior Member hercules's Avatar
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    Limit

    I'm having my dumb hour.

    lim (sin((4x+pi)/3)*sin(5x))/7x as x approaches zero

    Simplify and show all work

    we can use lim sin x/x -->1

    what is the final answer supposed to be?
    Thanks
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by hercules View Post
    I'm having my dumb hour.

    lim (sin((4x+pi)/3)*sin(5x))/7x as x approaches zero

    Simplify and show all work

    we can use lim sin x/x -->1

    what is the final answer supposed to be?
    Thanks
    \lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\cdot \frac{\sin(5x)}{7x}\right)

    We need to fix up the 2nd by making the denominator 5x

     <br />
\lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\right)\cdot \left(\lim_{x \to 0} \frac{5}{7} \left[ \frac{\sin(5x)}{5x}\right]\right)=(0)\cdot \left( \frac{5}{7}\left[1\right] \right)=0<br />
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  3. #3
    Junior Member hercules's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    \lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\cdot \frac{\sin(5x)}{7x}\right)

    We need to fix up the 2nd by making the denominator 5x

     <br />
\lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\right)\cdot \left(\lim_{x \to 0} \frac{5}{7} \left[ \frac{\sin(5x)}{5x}\right]\right)=(0)\cdot \left( \frac{5}{7}\left[1\right] \right)=0<br />
    Sorry, i guess i didn't write the problem clearly

    \lim_{x \to 0} \left( \frac{\sin((4x+\pi)/3)}{}\cdot \frac{\sin(5x)}{7x}\right)

    hope this is clearer (4x+pi)/3 is inside the sine
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Version 2.0 Hehe

    Okay here we go... The concept is the same.

    \lim_{x \to 0} \sin\left( \frac{(4x+\pi)}{3} \right) \cdot \left(\frac{\sin(5x)}{7x}\right)

    We need to fix up the 2nd by making the denominator 5x

     <br />
\lim_{x \to 0}\sin \left( \frac{(4x+\pi)}{3}\right)\cdot \left(\lim_{x \to 0} \frac{5}{7} \left[ \frac{\sin(5x)}{5x}\right]\right)=(\sin\left( \frac{\pi}{3}\right))\cdot \left( \frac{5}{7}\left[1\right] \right)=\left( \frac{\sqrt{3}}{2}\right)\cdot \left( \frac{5}{7}\right)=\frac{5\sqrt{3}}{14}<br />

    Good luck.
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by hercules View Post
    Sorry, i guess i didn't write the problem clearly

    \lim_{x \to 0} \left( \frac{\sin((4x+\pi)/3)}{}\cdot \frac{\sin(5x)}{7x}\right)

    hope this is clearer (4x+pi)/3 is inside the sine
    Then it would be:

    \lim_{x\to{0}} \left[\sin\bigg(\frac{4x+\pi}{3}\bigg)\cdot\frac{5}{7}\f  rac{\sin(5x)}{5x}\right]=\frac{\sqrt{3}}{2}\cdot\frac{5}{7}=\color{red}\bo  xed{\frac{5\sqrt{3}}{14}}
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