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Math Help - Limit of a Trig Function

  1. #1
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    Limit of a Trig Function

    I'm quite confused on this question.

    Limit Cos [ (pi * x) / (1- 4x) ]
    X -> infinity

    What I thought it was was to divide everything by x so it would be

    Cos (pi / (1/x) - 4)

    So as x -> inf 1/x = 0.
    So the answer would be

    Cos (pi / 4)

    am I doing this right?
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  2. #2
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    Quote Originally Posted by EliteNewbz View Post
    I'm quite confused on this question.

    Limit Cos [ (pi * x) / (1- 4x) ]
    X -> infinity

    What I thought it was was to divide everything by x so it would be

    Cos (pi / (1/x) - 4)

    So as x -> inf 1/x = 0.
    So the answer would be

    Cos (pi / 4)

    am I doing this right?
    Since the cosine function is continuous,

    \lim_{x \to \infty}\cos{\left(\frac{\pi x}{1 - 4x}\right)} = \cos{\left[\lim_{x \to \infty}\left(\frac{\pi x}{1 - 4x}\right)\right]}

    Since this goes to \frac{\phantom{-}\infty}{-\infty} we can use L'Hospital's Rule, so

    \cos{\left[\lim_{x \to \infty}\left(\frac{\pi x}{1 - 4x}\right)\right]} = \cos{\left[\lim_{x \to \infty}\left(\frac{\phantom{-}\pi}{-4}\right)\right]}

     = \cos{\left(-\frac{\pi}{4}\right)}

     = \cos{\frac{\pi}{4}}

     = \frac{\sqrt{2}}{2}.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by EliteNewbz View Post
    I'm quite confused on this question.

    Limit Cos [ (pi * x) / (1- 4x) ]
    X -> infinity

    What I thought it was was to divide everything by x so it would be

    Cos (pi / (1/x) - 4)

    So as x -> inf 1/x = 0.
    So the answer would be

    Cos (pi / 4)

    am I doing this right?
    Once you do what ProveIt does as a first step:

    As \cos is continuous:

    \lim_{x \to 0}\cos\left(\frac{\pi x}{1-4x}\right)=\cos\left[\lim_{x \to 0} \left(\frac{\pi x}{1-4x}\right)\right]

    yes.

    CB
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