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Evaluate the limit Squeeze Theorem Perhaps?

  1. Oct 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Evaluate the limit, if it exists:

    lim[itex](\frac{sin x}{4x} * \frac{5-5 cos 3x}{2})[/itex]
    x→0


    3. The attempt at a solution

    I understand that [itex]\frac{sin(x)}{4x}* \frac{4}{4} =\frac{1}{4}[/itex] but I don't know
    what to do next because the 5-5cos3x/2 trips me up. I'm not seeing anything that I can do to it, so I'm thinking the Squeeze Theorem. Any help progressing further would be appreciated.
     
  2. jcsd
  3. Oct 7, 2011 #2
    Start in pieces. Take the constants to the outside, and you're left with two terms.

    [tex]\lim_{x\rightarrow 0} \left (\frac{\sin x}{4x}\cdot\frac{5-5\cos 3x}{2} \right) = \frac{5}{8}\lim_{x\rightarrow 0} \left ( \frac{\sin x}{x} - \frac{\sin x \cos 3x}{x} \right)[/tex]

    See if you can take it from there.
     
  4. Oct 7, 2011 #3

    HallsofIvy

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    I wouldn't divide it up quite the way process91 does. Look at
    [tex]\frac{5}{8}\frac{sin x}{x}(1- cos(3x))[/tex]
    and it should be obvious.

    (If there had been an [itex]x^2[/itex] in the denominator rather than x, I would have made it
    [tex]\frac{5}{8}\frac{sin(x)}{x}(3)\frac{1- cos(3x)}{3x}[/tex]
    and it would be a bit more interesting!)
     
    Last edited: Oct 7, 2011
  5. Oct 7, 2011 #4

    Ray Vickson

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    You can use the fact that lim(f(x)*g(x)) = (lim f(x)) * (lim g(x)), if both limits on the right exist.

    RGV
     
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