Horizontal asymptotes of arctan(x)

Recall that $\arctan(x)$ is defined as the inverse of $\tan(\theta)$ for $\theta \in (-\pi/2, \pi/2)$, and its range is exactly this open interval. As $x \to +\infty$, the angle $\theta$ whose tangent equals $x$ must approach the upper boundary $\pi/2$, but never reach it since tangent is undefined there. This gives $\lim_{x \to +\infty} \arctan(x) = \pi/2$.

Similarly, as $x \to -\infty$, the angle approaches the lower boundary $-\pi/2$, so $\lim_{x \to -\infty} \arctan(x) = -\pi/2$. Since both limits are finite, each corresponds to a horizontal asymptote: $y = \pi/2$ as $x \to +\infty$ and $y = -\pi/2$ as $x \to -\infty$. Thus $g(x) = \arctan(x)$ has two distinct horizontal asymptotes.