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Math Help - Proving that a func converges

  1. #1
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    Proving that a func converges

    F(x)=cos(x)/x
    Can someone help me to prove that this function converges to +infinity when x->0+
    (With epsilon and delta)
    Last edited by davidciprut; December 31st 2013 at 03:51 AM.
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  2. #2
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    Re: Proving that a func converges

    You are misusing the term "converges". A function that tends to infinity diverges. I believe you are trying to show the limit exists as an infinity limit.

    You should know some properties of \cos x. For instance, you should know that \cos x is a strictly decreasing function on the interval (0,\pi). Since \cos 0 = 1\text{ and } \cos \dfrac{\pi}{3} = \dfrac{1}{2}, you know that \cos x > \dfrac{1}{2} for all x \in \left(0,\dfrac{\pi}{3}\right). Hence, \dfrac{1}{2x} < F(x) for all x \in \left(0,\dfrac{\pi}{3}\right). Now, given any \varepsilon>0, find 0<\delta < \dfrac{\pi}{3} such that x \in (0,\delta) implies \varepsilon < \dfrac{1}{2x} < F(x).
    Thanks from davidciprut
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  3. #3
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    Re: Proving that a func converges

    Intuitively, what is happening is this:

    cos(x)/x = (cos(x))(1/x).

    The first factor is going to be bounded (by 1 above, and some positive number > 0 below) on an interval (0,δ) if δ is small enough. The second factor is going to get very large if δ is small enough.

    So suppose we need to pick δ so that cos(x)/x > M, for some (arbitrarily large) integer M.

    Pick δ so that (at least), 1/δ > 2M (what does this mean? It means that δ < 1/(2M), of course).

    Now we need to perhaps further restrict δ so that cos(δ) > 1/2. As SlipEternal showed, this means choosing δ < pi/3 (there is nothing special about pi/3, it's just that we KNOW the value of cos(pi/3), and pi/3 lies in the interval (0,pi/2)...we could use ANY number in this interval, but coming up with a value for cos at this number might not be so much fun. pi/6 or pi/4 would work just as well, but the math would be "messier").

    So if δ < min(1/(2M),pi/3), we have:

    cos(x)/x > cos(x)/δ > cos(δ)/δ > (1/2)/(1/(2M)) = 1/(1/M) = M.

    (I state it this way, using M instead of ε, because ε is usually used for a very small number, and we want to show that cos(x)/x near 0 (but on the positive side) is very LARGE, in fact, if we get "close enough" we can make it larger than any positive integer M, so it must be infinite).
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