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.