Monday

December 22, 2014

December 22, 2014

Posted by **Frederique** on Sunday, November 19, 2006 at 9:36am.

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 :)

**Answer this Question**

**Related Questions**

calculus - Let f be a function defined by f(x)= arctan x/2 + arctan x. the value...

limiting position of the particle - A particle moves along the x axis so that ...

Calculus - Use continuity to evaluate. as x approaches 2, lim arctan((2x^2-8)/(...

math - i need some serious help with limits in pre-calc. here are a few ...

precal - The values of x that are solutions to the equation cos^(2)x=sin2x in ...

calculus - Now we prove Machin's formula using the tangent addition formula: tan...

Math - Arrange these in order from least to greatest: arctan(-sqrt3), arctan 0, ...

CALCULUS - Evaluate each of the following. (a) lim x->0(e^x)-1-x/ x^2 (b) lim...

calculus - just wondering if someoen could help me with this limit..: lim arctan...

Calculus - Find the following limits algebraically or explain why they don’t ...