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

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Answer 1 · 2016-02-15T11:20:46

#pi/2#

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#.