Determining trigonometric limit

Question

How can I determine the limit as $x \rightarrow \infty$ of:

$\tan^{-1} * e^x$

First of all, I can't conceptualize this...I didn't think a trigonometric function could have a limit because it is constantly changing.

I also just don't even know which steps to take. $e^\infty$ will be infinity, but what is $\frac\cos\sin$ of infinity?

Answer 1

I will assume you want to find $$\lim_{x\rightarrow\infty}e^x\tan^{-1} x$$ (since the question was not clearly stated, and I cannot edit since another edit is pending). $\tan^{-1} x$ will go toward $\frac{\pi}{2}$ (inverse trig is different than trig, and $\tan^{-1} x\not=\frac{1}{\tan x}$). $e^x$ goes toward $\infty$, so the limit is $\infty$.

Answer 2

If $\tan^{-1} x$ is $\arctan x$, then $e^\infty=\infty$ and $\arctan\infty=\pi/2$, so $$\lim_{x\to\infty}\tan^{-1}e^x=\frac{\pi}{2}$$