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Help Finding limit sinxcosx/x

  1. Aug 28, 2014 #1
    1. The problem statement, all variables and given/known data
    lim x->0 sinxcosx/x


    2. Relevant equations
    lim x->0 sinx/x = 1



    3. The attempt at a solution
    Pretty sure I need to use above property but I believe cosx/x is undef.
     
  2. jcsd
  3. Aug 28, 2014 #2

    pasmith

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    Recall the identity [itex]\sin(2x) = 2\sin x \cos x[/itex].
     
  4. Aug 28, 2014 #3

    LCKurtz

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    Or write it as$$
    \frac {\sin x} x \cdot \cos x$$
     
  5. Aug 28, 2014 #4
    limx->o sinx/x * limx->0 cosx = 1*1 =1
     
  6. Aug 29, 2014 #5
    Correct.
     
  7. Aug 29, 2014 #6

    HallsofIvy

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    That's the simplest way to do it but using pasmith's suggestion, since sin(2x)= 2sin(x)cos(x), sin(x)cos(x)= sin(2x)/x so that sin(x)cos(x)/x= sin(2x)/2x. Now let u= 2x. As x goes to 0, so does u= 2x and we have
    [tex]\lim_{x\to 0} \frac{sin(x)cos(x)}{x}= \lim_{u\to 0}\frac{sin(u)}{u}= 1[/tex].
     
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