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Math Help - Solving lim x->∞ e^x cos x

  1. #1
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    Question Solving lim x->∞ e^x cos x

    Hi everybody :)

    I'm trying to solve this limit


    \lim_{x\to\infty}e^x cos(x)


    Since \lim_{x\to\infty}e^x = \infty

    and \lim_{x\to\infty}cos(x) is a Real number in [-1,1]

    is it correct to say that the solution is \infty?

    Thank you for your help :)
    Last edited by gesoocreesto; June 10th 2013 at 04:39 PM.
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  2. #2
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    Re: Solving lim x->∞ e^x cos x

    Quote Originally Posted by gesoocreesto View Post
    to solve this limit
    \lim_{x\to\infty}e^x cos(x)

    Since \lim_{x\to\infty}e^x = \infty
    and \lim_{x\to\infty}cos(x) is a Real number in [-1,1]
    is it correct to say that the solution is \infty?
    Write f(x)=e^x\cos(x). Then f is continuous.

    Let for n\in\mathbb{Z}^+ define {a_n} = \frac{{\left( {n + 1} \right)\pi }}{2} is it true that \lim_{n\to\infty}f(a_n)=0~?

    Now what do you think about your question now?
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  3. #3
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    Re: Solving lim x->∞ e^x cos x

    No, it's not true. That limit goes to infinity. But I'm sorry I don't get the point
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  4. #4
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    Re: Solving lim x->∞ e^x cos x

    Quote Originally Posted by gesoocreesto View Post
    No, it's not true. That limit goes to infinity. But I'm sorry I don't get the point
    It certainly is true!

    You had better check you basic knowledge to trig values.

    For any odd multiple of \pi/2 the \cos is zero.
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  5. #5
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    Re: Solving lim x->∞ e^x cos x

    Got it, thank you for your help :)
    I ended up understanding that the solution I was searching for is based on the definition of limit superior/inferior. So

    \limsup\limits_{x\rightarrow\infty}e^x cos (x) = +\infty

    \liminf\limits_{x\rightarrow\infty}e^x cos (x) = -\infty
    Last edited by gesoocreesto; June 10th 2013 at 07:22 PM.
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