Hey,

I wanted to ask how is it possible to proof that ArcTan(1/x^2) limit when x goes to infinity, is 0 ?

I need a valid proof and not a intuitive proof (Epsilon Delta\ Or limit arithmetic ..)

Thanks a lot :)

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- Dec 7th 2012, 11:46 AMMichaelEngstlerProof ArcTan limit
Hey,

I wanted to ask how is it possible to proof that ArcTan(1/x^2) limit when x goes to infinity, is 0 ?

I need a valid proof and not a intuitive proof (Epsilon Delta\ Or limit arithmetic ..)

Thanks a lot :) - Dec 7th 2012, 12:14 PMHallsofIvyRe: Proof ArcTan limit
Well, arctangent is the inverse to tangent: y= arctan(1/x^2) is the same as 1/x^2= tan(y). I assume you know that as x goes to infinity, 1/x^2 goes to 0. So what must tan(y) be? And then, what must y be?

Now that's "intuitive" and not a "valid proof" but it is thinking like that that will lead you to a "valid proof". We want to show that in order to make arctan(1/x^2) as close to 0 as we please, , which is , , , .

So take and work backwards: if then . - Dec 7th 2012, 12:27 PMPlatoRe: Proof ArcTan limit
- Dec 8th 2012, 04:50 AMMichaelEngstlerRe: Proof ArcTan limit
Thanks both for your help,

It was very helpful ! (: - Dec 8th 2012, 05:16 AMMichaelEngstlerRe: Proof ArcTan limit
A question for

**HallsofIvy:**

On line 3, You got from 0<1/x^2<tan(e) to x^2>tan(e) .. shouldnt it be x^2 > 1/tan(e) ? And then the final requirement x>1/sqrt(tan(e)) ? - Dec 8th 2012, 05:25 AMHallsofIvyRe: Proof ArcTan limit
Yes, that was a typo. I intended to type cot(e).

- Dec 8th 2012, 05:30 AMMichaelEngstlerRe: Proof ArcTan limit
I started to write the proof and found another problem, Say e = 2.5.

I want N to be > 1/sqrt(tan(2.5)) .. But 1/sqrt(tan(2.5)) is undefined because tan(2.5) < 0. - Dec 8th 2012, 05:52 AMPlatoRe: Proof ArcTan limit