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

Feb 15, 2016

$\frac{\pi}{2}$

Explanation:

Let $y = {x}^{5} - \frac{1}{x} ^ 7$.

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

${\lim}_{x \to \infty} y = \infty$

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 $y = - \frac{\pi}{2}$ and $y = \frac{\pi}{2}$.

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

${\lim}_{x \to \infty} \arctan \left({x}^{5} - {x}^{- 7}\right) = {\lim}_{x \to \infty} \arctan \left(y\right)$

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

${\lim}_{x \to \infty} \arctan \left(y\right) = {\lim}_{y \to \infty} \arctan \left(y\right)$

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

${\lim}_{y \to \infty} \arctan \left(y\right)$

is simply $\frac{\pi}{2}$.