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Math Help - Limit of arctan(C tan(x))

  1. #1
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    Limit of arctan(C tan(x))

    Hello

    What is

    \mathrm{\lim_{x\rightarrow\frac{\pi}{2}}}\arctan\l  eft[C\tan(x)\right]

    where C is a constant. My guess is that it is -\frac{\pi}{2} for C < 0 and \frac{\pi}{2} for C > 0. However, if this is correct I require more technical treatment.

    Many thanks in advance.
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  2. #2
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    I evaluated this limit numerically and it obviously converges to \pi/2 when C > 0 and to - \pi/2 when C < 0 as x converges to \pi / 2. However, I am still looking for a formal proof.
    Thanks.
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  3. #3
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    Quote Originally Posted by JulieK View Post
    Hello

    What is

    \mathrm{\lim_{x\rightarrow\frac{\pi}{2}}}\arctan\l  eft[C\tan(x)\right]

    where C is a constant. My guess is that it is -\frac{\pi}{2} for C < 0 and \frac{\pi}{2} for C > 0. However, if this is correct I require more technical treatment.

    Many thanks in advance.
    Dear JulieK,

    The limit does not exist.

    Notice that, \lim_{x\rightarrow{\frac{\pi}{2}}^+}(\tan{x})\neq{  \lim_{x\rightarrow{\frac{\pi}{2}}^{-}}(\tan{x})}
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  4. #4
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    Dear Sudharaka

    Thank you for your reply. Yes, you are right. But what about one-sided limits. In fact I am mainly interested in \lim_{x\rightarrow{\frac{\pi}{2}}^-}(\arctan(C\tan{x})) and that is what I numerically evaluated.
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  5. #5
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    Quote Originally Posted by JulieK View Post
    Dear Sudharaka

    Thank you for your reply. Yes, you are right. But what about one-sided limits. In fact I am mainly interested in \lim_{x\rightarrow{\frac{\pi}{2}}^-}(\arctan(C\tan{x})) and that is what I numerically evaluated.
    Dear JulieK,

    \lim_{x\rightarrow{\frac{\pi}{2}}^-}tan^{-1}(C\tan{x})=tan^{-1}\left[c\lim_{x\rightarrow{\frac{\pi}{2}}^-}\tan{x}\right] ; Since tan^{-1} is continuous.

    If~c>0\Rightarrow~y=c\lim_{x\rightarrow{\frac{\pi}  {2}}^-}\tan{x}=\infty

    Therefore, \lim_{x\rightarrow{\frac{\pi}{2}}^-}tan^{-1}(C\tan{x})=\lim_{y\rightarrow\infty}tan^{-1}y=\frac{\pi}{2}

    If~c<0\Rightarrow~y=c\lim_{x\rightarrow{\frac{\pi}  {2}}^-}\tan{x}=-\infty

    Therefore, \lim_{x\rightarrow{\frac{\pi}{2}}^-}tan^{-1}(C\tan{x})=\lim_{y\rightarrow~-\infty}tan^{-1}y=-\frac{\pi}{2}

    Hope this will help you.
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  6. #6
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    Dear Sudharaka
    Thank you very much.
    Best wishes
    JulieK
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  7. #7
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    Quote Originally Posted by JulieK View Post
    Dear Sudharaka
    Thank you very much.
    Best wishes
    JulieK
    Dear JulieK,

    You are welcome.
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