Question

How do you find the limit of `arctan (x^5-x^-7)` as x approaches infinity?

Answer 1

`pi/2`

Explanation:

Let `y = x^5 - 1/x^7`.

When `x` keeps increasing, the second term of `y` slowly vanishes. So, it should be clear that

`lim_{x -> oo} y = oo`

Now, to solve the original question, take a look at the `arctan` graph first.

graph{arctan(x) [-10, 10, -5, 5]}

Notice that there are `2` horizontal asymptotes, namely `y = -pi/2` and `y = pi/2`.

To find the limit, substitute the interior of the `arctan` with `y`. So it becomes

`lim_{x -> oo} arctan(x^5 - x^{-7}) = lim_{x -> oo} arctan(y)`

And previously, we know that as `x` tends to infinity, so does `y`. Therefore we can write

`lim_{x -> oo} arctan(y) = lim_{y -> oo} arctan(y)`

And if you look at the graph above one more time, you can see that

`lim_{y -> oo} arctan(y)`

is simply `pi/2`.