# Thread: Limit of arctan(C tan(x))

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

Hello

What is

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

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

Many thanks in advance.

2. I evaluated this limit numerically and it obviously converges to $\displaystyle \pi/2$ when $\displaystyle C > 0$ and to $\displaystyle - \pi/2$ when $\displaystyle C < 0$ as $\displaystyle x$ converges to $\displaystyle \pi / 2$. However, I am still looking for a formal proof.
Thanks.

3. Originally Posted by JulieK
Hello

What is

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

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

Many thanks in advance.
Dear JulieK,

The limit does not exist.

Notice that, $\displaystyle \lim_{x\rightarrow{\frac{\pi}{2}}^+}(\tan{x})\neq{ \lim_{x\rightarrow{\frac{\pi}{2}}^{-}}(\tan{x})}$

4. Dear Sudharaka

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

5. Originally Posted by JulieK
Dear Sudharaka

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

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

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

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

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

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

6. Dear Sudharaka
Thank you very much.
Best wishes
JulieK

7. Originally Posted by JulieK
Dear Sudharaka
Thank you very much.
Best wishes
JulieK
Dear JulieK,

You are welcome.