# How do you find the limit of  arctan(x)  as x approaches infinity?

Jul 4, 2016

The limit is $\frac{\pi}{2}$ against the principal-value convention for $y = a r c \tan x .$
General limit $\left(2 n + \frac{1}{2}\right) \pi , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$,

#### Explanation:

For the principal value $y = a r c \tan x$, as $x \to \infty , y \to \frac{\pi}{2}$

The general limit

= the principal-value limit $+ n \pi , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$.

So, the general limit is $\frac{\pi}{2} +$even multiple of pi$, a s$ x to oo#

and $\frac{\pi}{2} +$odd multiple of $\pi$, as x $\to - \infty$.