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Thread: 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


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


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

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

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

    Thank you for your help :)
    Last edited by gesoocreesto; Jun 10th 2013 at 03:39 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
    $\displaystyle \lim_{x\to\infty}e^x cos(x)$

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

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

    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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    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 $\displaystyle \pi/2$ the $\displaystyle \cos$ 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

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

    $\displaystyle \liminf\limits_{x\rightarrow\infty}e^x cos (x) = -\infty$
    Last edited by gesoocreesto; Jun 10th 2013 at 06:22 PM.
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