If one is undefined, shouldn't the other be undefined, too? They are inverse functions. For instance, since we have that
$$\lim_{x \to \frac{\pi}{2}} \tan x = \text{undefined}$$
so too should $$\lim_{x \to \infty} \arctan x = \text{undefined}$$
since there is no value for which $\tan x$ takes infinity.
Best Answer
The limit of tangent at $\frac\pi2$ is undefined, but the limits approaching from below and above are defined: $$\lim_{x\rightarrow\frac\pi2,\;x<\frac\pi2}\tan(x)=+\infty, \\ \lim_{x\rightarrow\frac\pi2,\;x>\frac\pi2}\tan(x)=-\infty.$$ In particular, if we restrict the function $\tan$ to the open interval $(-\frac\pi2,\frac\pi2)$, then $\tan$ has limits at both ends.
The important thing to know is that $\arctan$ is defined as the inverse of this restricted function, not the general function on $\Bbb R$ (which is not even bijective). Therefore, nothing prevents $\arctan$ from having limits at $-\infty$ and $+\infty$, and in fact $$\lim_{x\rightarrow+\infty}\arctan(x)=\frac\pi2, \\ \lim_{x\rightarrow-\infty}\arctan(x)=-\frac\pi2.$$