# Proof ArcTan limit

• December 7th 2012, 11:46 AM
MichaelEngstler
Proof 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 :)
• December 7th 2012, 12:14 PM
HallsofIvy
Re: 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, $|arctan(1/x^2)|< \epsilon$, which is $-\epsilon< arctan(1/x^2)< \epsilon$, $0 < 1/x^2< tan(\epsilon)$, $x^2> tan(\epsilon)$, $x> \sqrt{tan(\epsilon)$.

So take $N> \sqrt{tan(\epsilon)}$ and work backwards: if $x> N$ then $|arctan(1/x^2)|< \epsilon$.
• December 7th 2012, 12:27 PM
Plato
Re: Proof ArcTan limit
Quote:

Originally Posted by MichaelEngstler
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 ..)

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

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

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

And $\left( {\forall \delta > 0} \right)\left( {\exists N \in \mathbb{N}} \right)\left[{x \geqslant N \to \frac{1}{x} < \delta } \right]$
• December 8th 2012, 04:50 AM
MichaelEngstler
Re: Proof ArcTan limit
It was very helpful ! (:
• December 8th 2012, 05:16 AM
MichaelEngstler
Re: 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)) ?
• December 8th 2012, 05:25 AM
HallsofIvy
Re: Proof ArcTan limit
Yes, that was a typo. I intended to type cot(e).
• December 8th 2012, 05:30 AM
MichaelEngstler
Re: 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.
• December 8th 2012, 05:52 AM
Plato
Re: Proof ArcTan limit
Quote:

Originally Posted by MichaelEngstler
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.

If you work only with $\arctan(t)$ you will avoid those kinds of problems.