Question

How do you find the limit of `arctan(x)` as x approaches `oo`?

Answer 1

Think about the tangent function.

Explanation:

Recall that `arctan x` is a number (angle) `theta` in `(-pi/2,pi/2)` with `tan theta = x`.

To answer the question you need to figure out:
What does `theta` have to get close to for the tangent to get greater and greater without any bound on how big it gets?

Note that `tan 0 =0`, but as `theta` increases, so does the tangent.

In fact as `theta` gets closer to `pi/2` the tangent increases without bound.

So if we increase the tangent without bound, then the corresponding angle (number) approaches `pi/2`

`lim_(xrarroo) arctan x = pi/2`