Note that

pi
lim arctan(x ) = ----
x -> +oo 2

Now evaluate

/ pi \
lim |arctan(x ) - -----| x
x -> +oo \ 2 /

I'm not exactly sure how to attempt it. I have tried h'opital's rule but I don't believe you can use it here. Any help will be greatly appreciated!

Sorry the question came out weirder than i had originally posted it

it is the lim as x approaches positive infinity of (arctanx - pi/2)x

Use that:

arctan(x) = pi/2 - arctan(1/x)

If you take a right triangle then you can easily see where this relation comes from. If x is the ratio between two right sides then 1/x is the inverse of that ratio, so arctan(1/x) will yield the other angle which is pi minus arctan(x).

If x approaches infinity, 1/x approaches zero, so you can use the series expansion of the arctangent function around x = 0:

arctan(x) = x - x^3/3 + x^5/5 -...
for x in a neighborhood of zero --->

arctan(1/x) = x^(-1) - x^(-3)/3 + x^(-5)/5 -..

for x -->infinity

Therefore for large x:

arctan(x) = pi/2 - arctan(1/x)=

pi/2 - 1/x + 1/(3 x^3) - ...

And you can now read-off the desired limit :)

To evaluate the given limit, we can use the formula for arctan(x) and the series expansion of the arctangent function.

First, note that arctan(x) can be expressed as:

arctan(x) = π/2 - arctan(1/x)

This identity can be derived from a right triangle where x is the ratio between two right sides and 1/x is the inverse of that ratio. arctan(1/x) will yield the other angle in the triangle, which is π - arctan(x).

Next, we can use the series expansion of the arctangent function around x = 0. The series expansion of arctan(x) is:

arctan(x) = x - x^3/3 + x^5/5 - ...

This expansion holds for x in a neighborhood of zero.

Similarly, the series expansion of arctan(1/x) is:

arctan(1/x) = 1/x - (1/x)^3/3 + (1/x)^5/5 - ...

Since x approaches positive infinity, 1/x approaches zero. Therefore, we can simplify the expression for arctan(x) using the series expansion:

arctan(x) = π/2 - 1/x + 1/(3x^3) - ...

Now, we can evaluate the limit as x approaches positive infinity:

lim(x -> ∞) (arctan(x) - π/2)x

Substituting the expression for arctan(x), we have:

lim(x -> ∞) ((π/2 - 1/x + 1/(3x^3) - π/2)x

Now, we can expand the expression and simplify:

lim(x -> ∞) (πx/2 - x/x + x/(3x^3) - πx/2)

lim(x -> ∞) (0 + 0 + 1/(3x^2) - 0)

lim(x -> ∞) 1/(3x^2)

As x approaches positive infinity, the term 1/(3x^2) approaches zero. Therefore, the value of the limit is zero.

In summary, the limit as x approaches positive infinity of (arctan(x) - π/2)x is equal to zero.