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Math Help - Find limit

  1. #1
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    Find limit

    \lim_{x \to 0} cos(pi/x)/(x - 2)

    How could I find this limit? Any idea would be appreciable.
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  2. #2
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    Re: Find limit

    Quote Originally Posted by initM View Post
    \lim_{x \to 0} cos(pi/x)/(x - 2)

    How could I find this limit? Any idea would be appreciable.
    The limit does not exist, because for values infinitessimally close to \displaystyle x = 0, the function \displaystyle \cos{\left(\frac{\pi}{x}\right)} can equal any value in \displaystyle [-1, 1] (you can not determine a single value that it goes to).
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Find limit

    Quote Originally Posted by initM View Post
    \lim_{x \to 0} cos(pi/x)/(x - 2)

    How could I find this limit? Any idea would be appreciable.
    Take x_n=\frac{1}{2\pi n} and y_n=\frac{1}{(2n+1)\pi} and x_n ,y_n \to 0 when n\to \infty.

    Could you continue?
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  4. #4
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    Re: Find limit

    Quote Originally Posted by Also sprach Zarathustra View Post
    Take x_n=\frac{1}{2\pi n} and y_n=\frac{1}{(2n+1)\pi} and x_n ,y_n \to 0 when n\to \infty.

    Could you continue?
    It is little bit difficult for me. Can I use any substitution here?
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  5. #5
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    Re: Find limit

    Quote Originally Posted by initM View Post
    It is little bit difficult for me. Can I use any substitution here?
    Actually, I think there is a slight error there- the " \pi" should not be in there. Try instead x_n= \frac{1}{2n} and y_n= \frac{1}{2n+1}, 1 over the even integers and 1 over the odd integers.

    Now, I am hoping that "It is a little bit difficult for me" does NOT mean "I really don't want to actually do anything my self" so I will make some suggestions: If x_n= \frac{1}{2n}, what is \frac{\pi}{x_n}? What is the cosine of that? What does that go to as n goes to infinity? If y_n= \frac{1}{2n+1}, what is \frac{\pi}{y_n}? What is the cosine of that? What does that go to as n goes to infinity?
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