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, $\displaystyle |arctan(1/x^2)|< \epsilon$, which is $\displaystyle -\epsilon< arctan(1/x^2)< \epsilon$, $\displaystyle 0 < 1/x^2< tan(\epsilon)$, $\displaystyle x^2> tan(\epsilon)$, $\displaystyle x> \sqrt{tan(\epsilon)$.

So take $\displaystyle N> \sqrt{tan(\epsilon)}$ and work backwards: if $\displaystyle x> N$ then $\displaystyle |arctan(1/x^2)|< \epsilon$. - Dec 7th 2012, 12:27 PMPlatoRe: Proof ArcTan limit

You know that $\displaystyle \arctan(x)$ is a continuous function.

And $\displaystyle \arctan(0)=0$ also $\displaystyle \lim _{x \to \infty } \frac{1}{x} = 0$.

Combine these $\displaystyle \left( {\forall \varepsilon > 0} \right)\left( {\exists \delta > 0} \right)\left[ {\left| x \right| < \delta \to \left| {\arctan (x)} \right| < \varepsilon } \right]$.

And $\displaystyle \left( {\forall \delta > 0} \right)\left( {\exists N \in \mathbb{N}} \right)\left[{x \geqslant N \to \frac{1}{x} < \delta } \right]$ - 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